(i) A⊗RK≅K[X1,1X1,…,Xn,1Xn]A⊗RK≅KX1,1X1,…,Xn,1XnAox_(R)K~=K[X_(1),(1)/(X_(1)),dots,X_(n),(1)/(X_(n))]A \otimes_{R} K \cong K\left[X_{1}, \frac{1}{X_{1}}, \ldots, X_{n}, \frac{1}{X_{n}}\right]A⊗RK≅K[X1,1X1,…,Xn,1Xn],
(ii) for each height-one prime ideal PPPPP of R,A⊗Rk(P)≅k(P)[X1,1X1,…R,A⊗Rk(P)≅k(P)X1,1X1,…R,Aox_(R)k(P)~=k(P)[X_(1),(1)/(X_(1)),dots:}R, A \otimes_{R} k(P) \cong k(P)\left[X_{1}, \frac{1}{X_{1}}, \ldots\right.R,A⊗Rk(P)≅k(P)[X1,1X1,…, Xn,1Xn]Xn,1Xn{:X_(n),(1)/(X_(n))]\left.X_{n}, \frac{1}{X_{n}}\right]Xn,1Xn]
Then AAAAA is a locally Laurent polynomial algebra in nnnnn variables over RRRRR, i.e.,
and is of the form B[I−1]BI−1B[I^(-1)]B\left[I^{-1}\right]B[I−1], where BBBBB is the symmetric algebra of a projective RRRRR-module QQQQQ of rank n,Qn,Qn,Qn, Qn,Q is a direct sum of finitely generated projective RRRRR-modules of rank one, and IIIII is an invertible ideal of BBBBB.
5. EPIMORPHISM PROBLEM
The Epimorphism Problem for hypersurfaces asks the following fundamental question:
Question 4. Let kkkkk be a field and f∈B=k[n]f∈B=k[n]f in B=k^([n])f \in B=k^{[n]}f∈B=k[n] for some integer n≥2n≥2n >= 2n \geq 2n≥2. Suppose
Does this imply that B=k[f][n−1]B=k[f][n−1]B=k[f]^([n-1])B=k[f]^{[n-1]}B=k[f][n−1], i.e., is fffff a coordinate in BBBBB ?
This problem is generally known as the Epimorphism Problem. It is an open problem and is regarded as one of the most challenging and celebrated problems in the area of affine algebraic geometry (see [38,69,75,77][38,69,75,77][38,69,75,77][38,69,75,77][38,69,75,77] for useful surveys).
The first major breakthrough on Question 4 was achieved during 1974-1975, independently, by Abhyankar-Moh and Suzuki [5,86]. They showed that Question 4 has an affirmative answer when kkkkk is a field of characteristic zero and n=2n=2n=2n=2n=2. Over a field of positive characteristic, explicit examples of nonrectifiable epimorphisms from k[X,Y]k[X,Y]k[X,Y]k[X, Y]k[X,Y] to k[T]k[T]k[T]k[T]k[T] (referred to in Section 2) and hence explicit examples of nontrivial lines had already been demonstrated by Segre [83] in 1957 and Nagata [71] in 1971. However, over a field of characteristic zero, we have the following conjecture:
Abhyankar-Sathaye Conjecture. Let kkkkk be a field of characteristic zero and f∈B=k[n]f∈B=k[n]f in B=k^([n])f \in B=k^{[n]}f∈B=k[n] for some integer n≥2n≥2n >= 2n \geq 2n≥2. Suppose that B/(f)≅k[n−1]B/(f)≅k[n−1]B//(f)~=k^([n-1])B /(f) \cong k^{[n-1]}B/(f)≅k[n−1]. Then B=k[f][n−1]B=k[f][n−1]B=k[f]^([n-1])B=k[f]^{[n-1]}B=k[f][n−1].
In case n=3n=3n=3n=3n=3, some special cases have been solved by Sathaye, Russell, and Wright [73,76,79,89]. In [79], Sathaye proved the conjecture for the linear planes, i.e., polynomials FFFFF of the form aZ−baZ−baZ-ba Z-baZ−b, where a,b∈k[X,Y]a,b∈k[X,Y]a,b in k[X,Y]a, b \in k[X, Y]a,b∈k[X,Y]. This was further extended by Russell over fields of any characteristic. They proved that
Theorem 5.1. Let F∈k[X,Y,Z]F∈k[X,Y,Z]F in k[X,Y,Z]F \in k[X, Y, Z]F∈k[X,Y,Z] be such that F=aZ−bF=aZ−bF=aZ-bF=a Z-bF=aZ−b, where a(≠0),b∈k[X,Y]a(≠0),b∈k[X,Y]a(!=0),b in k[X,Y]a(\neq 0), b \in k[X, Y]a(≠0),b∈k[X,Y], and k[X,Y,Z]/(F)=k[2]k[X,Y,Z]/(F)=k[2]k[X,Y,Z]//(F)=k^([2])k[X, Y, Z] /(F)=k^{[2]}k[X,Y,Z]/(F)=k[2]. Then there exist X0,Y0∈k[X,Y]X0,Y0∈k[X,Y]X_(0),Y_(0)in k[X,Y]X_{0}, Y_{0} \in k[X, Y]X0,Y0∈k[X,Y] such that k[X,Y]=k[X0,Y0]k[X,Y]=kX0,Y0k[X,Y]=k[X_(0),Y_(0)]k[X, Y]=k\left[X_{0}, Y_{0}\right]k[X,Y]=k[X0,Y0] with a∈k[X0]a∈kX0a in k[X_(0)]a \in k\left[X_{0}\right]a∈k[X0] and k[X,Y,Z]=k[X0,F][1]k[X,Y,Z]=kX0,F[1]k[X,Y,Z]=k[X_(0),F]^([1])k[X, Y, Z]=k\left[X_{0}, F\right]^{[1]}k[X,Y,Z]=k[X0,F][1].
When kkkkk is an algebraically closed field of characteristic p≥0p≥0p >= 0p \geq 0p≥0, Wright [89] proved the conjecture for polynomials FFFFF of the form aZm−baZm−baZ^(m)-ba Z^{m}-baZm−b with a,b∈k[X,Y],m≥2a,b∈k[X,Y],m≥2a,b in k[X,Y],m >= 2a, b \in k[X, Y], m \geq 2a,b∈k[X,Y],m≥2 and p∤mp∤mp∤mp \nmid mp∤m. Das and Dutta showed [28, THEOREM 4.5] that Wright's result extends to any field kkkkk. They proved that
Theorem 5.2. Let kkkkk be any field with ch k=p(≥0)k=p(≥0)k=p( >= 0)k=p(\geq 0)k=p(≥0) and F=aZm−b∈k[X,Y,Z]F=aZm−b∈k[X,Y,Z]F=aZ^(m)-b in k[X,Y,Z]F=a Z^{m}-b \in k[X, Y, Z]F=aZm−b∈k[X,Y,Z] be such that a(≠0),b∈k[X,Y],m≥2a(≠0),b∈k[X,Y],m≥2a(!=0),b in k[X,Y],m >= 2a(\neq 0), b \in k[X, Y], m \geq 2a(≠0),b∈k[X,Y],m≥2 and p∤mp∤mp∤mp \nmid mp∤m. Suppose that k[X,Y,Z]/(F)=k[2]k[X,Y,Z]/(F)=k[2]k[X,Y,Z]//(F)=k^([2])k[X, Y, Z] /(F)=k^{[2]}k[X,Y,Z]/(F)=k[2]. Then there exists X0∈k[X,Y]X0∈k[X,Y]X_(0)in k[X,Y]X_{0} \in k[X, Y]X0∈k[X,Y] such that k[X,Y]=k[X0,b]k[X,Y]=kX0,bk[X,Y]=k[X_(0),b]k[X, Y]=k\left[X_{0}, b\right]k[X,Y]=k[X0,b] with a∈k[X0]a∈kX0a in k[X_(0)]a \in k\left[X_{0}\right]a∈k[X0] and k[X,Y,Z]=k[X,Y,Z]=k[X,Y,Z]=k[X, Y, Z]=k[X,Y,Z]=k[F,Z,X0]kF,Z,X0k[F,Z,X_(0)]k\left[F, Z, X_{0}\right]k[F,Z,X0].
The condition that p∤mp∤mp∤mp \nmid mp∤m is necessary in Theorem 5.2 (cf. [28, REMARK 4.6]).
Most of the above cases are covered by the following generalization due to Russell and Sathaye [76, THEOREM 3.6]:
Theorem 5.3. Let kkkkk be a field of characteristic zero and let
F=amZm+am−1Zm−1+⋯+a1Z+a0∈k[X,Y,Z]F=amZm+am−1Zm−1+⋯+a1Z+a0∈k[X,Y,Z]F=a_(m)Z^(m)+a_(m-1)Z^(m-1)+cdots+a_(1)Z+a_(0)in k[X,Y,Z]F=a_{m} Z^{m}+a_{m-1} Z^{m-1}+\cdots+a_{1} Z+a_{0} \in k[X, Y, Z]F=amZm+am−1Zm−1+⋯+a1Z+a0∈k[X,Y,Z]
where a0,…,am∈k[X,Y]a0,…,am∈k[X,Y]a_(0),dots,a_(m)in k[X,Y]a_{0}, \ldots, a_{m} \in k[X, Y]a0,…,am∈k[X,Y] are such that GCD(a1,…,am)∉kGCDâ¡a1,…,am∉kGCD(a_(1),dots,a_(m))!in k\operatorname{GCD}\left(a_{1}, \ldots, a_{m}\right) \notin kGCDâ¡(a1,…,am)∉k. Suppose that
k[X,Y,Z]/(F)=k[2]k[X,Y,Z]/(F)=k[2]k[X,Y,Z]//(F)=k^([2])k[X, Y, Z] /(F)=k^{[2]}k[X,Y,Z]/(F)=k[2]
Then there exists X0∈k[X,Y]X0∈k[X,Y]X_(0)in k[X,Y]X_{0} \in k[X, Y]X0∈k[X,Y] such that k[X,Y]=k[X0,b]k[X,Y]=kX0,bk[X,Y]=k[X_(0),b]k[X, Y]=k\left[X_{0}, b\right]k[X,Y]=k[X0,b] with am∈k[X0]am∈kX0a_(m)in k[X_(0)]a_{m} \in k\left[X_{0}\right]am∈k[X0]. Further, k[X,Y,Z]=k[F][2]k[X,Y,Z]=k[F][2]k[X,Y,Z]=k[F]^([2])k[X, Y, Z]=k[F]^{[2]}k[X,Y,Z]=k[F][2].
Thus, for k[X,Y,Z]k[X,Y,Z]k[X,Y,Z]k[X, Y, Z]k[X,Y,Z], the Abhyankar-Sathaye conjecture remains open for the case when GCD(a1,…,am)=1GCDâ¡a1,…,am=1GCD(a_(1),dots,a_(m))=1\operatorname{GCD}\left(a_{1}, \ldots, a_{m}\right)=1GCDâ¡(a1,…,am)=1.
A common theme in most of the partial results proved in the Abhyankar-Sathaye conjecture for k[X,Y,Z]k[X,Y,Z]k[X,Y,Z]k[X, Y, Z]k[X,Y,Z] is that, if FFFFF is considered as a polynomial in ZZZZZ, then the coordinates of k[X,Y]k[X,Y]k[X,Y]k[X, Y]k[X,Y] can be so chosen that the coefficient of ZZZZZ becomes a polynomial in XXXXX. The Abhyankar-Sathaye conjecture for k[X,Y,Z]k[X,Y,Z]k[X,Y,Z]k[X, Y, Z]k[X,Y,Z] can now be split into two parts.
Question 4A. Let kkkkk be a field of characteristic zero and let
F=amZm+am−1Zm−1+⋯+a1Z+a0∈k[X,Y,Z]F=amZm+am−1Zm−1+⋯+a1Z+a0∈k[X,Y,Z]F=a_(m)Z^(m)+a_(m-1)Z^(m-1)+cdots+a_(1)Z+a_(0)in k[X,Y,Z]F=a_{m} Z^{m}+a_{m-1} Z^{m-1}+\cdots+a_{1} Z+a_{0} \in k[X, Y, Z]F=amZm+am−1Zm−1+⋯+a1Z+a0∈k[X,Y,Z]
where a0,…,am∈k[X,Y]a0,…,am∈k[X,Y]a_(0),dots,a_(m)in k[X,Y]a_{0}, \ldots, a_{m} \in k[X, Y]a0,…,am∈k[X,Y]. Suppose that k[X,Y,Z]/(F)=k[2]k[X,Y,Z]/(F)=k[2]k[X,Y,Z]//(F)=k^([2])k[X, Y, Z] /(F)=k^{[2]}k[X,Y,Z]/(F)=k[2]. Does there exist X0∈k[X,Y]X0∈k[X,Y]X_(0)in k[X,Y]X_{0} \in k[X, Y]X0∈k[X,Y] such that k[X,Y]=k[X0][1]k[X,Y]=kX0[1]k[X,Y]=k[X_(0)]^([1])k[X, Y]=k\left[X_{0}\right]^{[1]}k[X,Y]=k[X0][1] with am∈k[X0]am∈kX0a_(m)in k[X_(0)]a_{m} \in k\left[X_{0}\right]am∈k[X0] ?
Question 4B. Let kkkkk be a field of characteristic zero and suppose
F=am(X)Zm+am−1Zm−1+⋯+a1Z+a0∈k[X,Y,Z]F=am(X)Zm+am−1Zm−1+⋯+a1Z+a0∈k[X,Y,Z]F=a_(m)(X)Z^(m)+a_(m-1)Z^(m-1)+cdots+a_(1)Z+a_(0)in k[X,Y,Z]F=a_{m}(X) Z^{m}+a_{m-1} Z^{m-1}+\cdots+a_{1} Z+a_{0} \in k[X, Y, Z]F=am(X)Zm+am−1Zm−1+⋯+a1Z+a0∈k[X,Y,Z]
where a0,…,am−1∈k[X,Y]a0,…,am−1∈k[X,Y]a_(0),dots,a_(m-1)in k[X,Y]a_{0}, \ldots, a_{m-1} \in k[X, Y]a0,…,am−1∈k[X,Y] and am∈k[X]am∈k[X]a_(m)in k[X]a_{m} \in k[X]am∈k[X]. Suppose that k[X,Y,Z]/(F)=k[2]k[X,Y,Z]/(F)=k[2]k[X,Y,Z]//(F)=k^([2])k[X, Y, Z] /(F)=k^{[2]}k[X,Y,Z]/(F)=k[2]. Does this imply that k[X,Y,Z]=k[F][2]k[X,Y,Z]=k[F][2]k[X,Y,Z]=k[F]^([2])k[X, Y, Z]=k[F]^{[2]}k[X,Y,Z]=k[F][2] ?
Sangines Garcia in his PhD thesis [78] answered Question 4A affirmatively for the case m=2m=2m=2m=2m=2. In [21], Bhatwadekar and the author have given an alternative proof of this result of Garcia.
When kkkkk is any field, as a partial generalization of Theorem 5.1 and Question 4B in four variables, the author proved the Abhyankar-Sathaye conjecture for a polynomial FFFFF of the form XmY−F(X,Z,T)∈k[X,Y,Z,T]XmY−F(X,Z,T)∈k[X,Y,Z,T]X^(m)Y-F(X,Z,T)in k[X,Y,Z,T]X^{m} Y-F(X, Z, T) \in k[X, Y, Z, T]XmY−F(X,Z,T)∈k[X,Y,Z,T]. This was one of the consequences of her general investigation on the ZCP [46]. In the process, she related it with other central problems on affine spaces like the affine fibration problem and the ZCP. The author has proved equivalence of ten statements, some of which involve an invariant introduced by Derksen, which is called the Derksen invariant.
The Derksen invariant of an integral domain BBBBB, denoted by DK(B)DK(B)DK(B)\mathrm{DK}(B)DK(B), is defined as the smallest subring of BBBBB generated by the kernel of DDDDD, where DDDDD varies over the set of all locally nilpotent derivations of BBBBB.
Theorem 5.4. Let kkkkk be a field of any characteristic and AAAAA an integral domain defined by
A=k[X,Y,Z,T]/(XmY−F(X,Z,T)), where m>1A=k[X,Y,Z,T]/XmY−F(X,Z,T), where m>1A=k[X,Y,Z,T]//(X^(m)Y-F(X,Z,T)),quad" where "m > 1A=k[X, Y, Z, T] /\left(X^{m} Y-F(X, Z, T)\right), \quad \text { where } m>1A=k[X,Y,Z,T]/(XmY−F(X,Z,T)), where m>1
Let x,y,zx,y,zx,y,zx, y, zx,y,z, and ttttt denote, respectively, the images of X,Y,ZX,Y,ZX,Y,ZX, Y, ZX,Y,Z, and TTTTT in AAAAA. Set f(Z,T):=f(Z,T):=f(Z,T):=f(Z, T):=f(Z,T):=F(0,Z,T)F(0,Z,T)F(0,Z,T)F(0, Z, T)F(0,Z,T) and G:=XmY−F(X,Z,T)G:=XmY−F(X,Z,T)G:=X^(m)Y-F(X,Z,T)G:=X^{m} Y-F(X, Z, T)G:=XmY−F(X,Z,T). Then the following statements are equivalent:
(i) k[X,Y,Z,T]=k[X,G][2]k[X,Y,Z,T]=k[X,G][2]k[X,Y,Z,T]=k[X,G]^([2])k[X, Y, Z, T]=k[X, G]^{[2]}k[X,Y,Z,T]=k[X,G][2].
(ii) k[X,Y,Z,T]=k[G][3]k[X,Y,Z,T]=k[G][3]k[X,Y,Z,T]=k[G]^([3])k[X, Y, Z, T]=k[G]^{[3]}k[X,Y,Z,T]=k[G][3].
(v) A[ℓ]≅kk[ℓ+3]A[â„“]≅kk[â„“+3]quadA^([â„“])~=_(k)k^([â„“+3])\quad A^{[\ell]} \cong_{k} k^{[\ell+3]}A[â„“]≅kk[â„“+3] for some integer ℓ≥0ℓ≥0â„“ >= 0\ell \geq 0ℓ≥0 and DK(A)≠k[x,z,t]DK(A)≠k[x,z,t]DK(A)!=k[x,z,t]\mathrm{DK}(A) \neq k[x, z, t]DK(A)≠k[x,z,t].
(vi) AAquad A\quad AA is an A2A2A^(2)\mathbb{A}^{2}A2-fibration over k[x]k[x]k[x]k[x]k[x] and DK(A)≠k[x,z,t]DK(A)≠k[x,z,t]DK(A)!=k[x,z,t]\mathrm{DK}(A) \neq k[x, z, t]DK(A)≠k[x,z,t].
(vii) AAAAA is geometrically factorial over k,DK(A)≠k[x,z,t]k,DKâ¡(A)≠k[x,z,t]k,DK(A)!=k[x,z,t]k, \operatorname{DK}(A) \neq k[x, z, t]k,DKâ¡(A)≠k[x,z,t] and the canonical map k∗→K1(A)k∗→K1(A)k^(**)rarrK_(1)(A)k^{*} \rightarrow K_{1}(A)k∗→K1(A) (induced by the inclusion k↪Ak↪Ak↪Ak \hookrightarrow Ak↪A ) is an isomorphism.
(viii) A is geometrically factorial over k,DK(A)≠k[x,z,t]k,DKâ¡(A)≠k[x,z,t]k,DK(A)!=k[x,z,t]k, \operatorname{DK}(A) \neq k[x, z, t]k,DKâ¡(A)≠k[x,z,t] and (A/xA)∗=k∗(A/xA)∗=k∗(A//xA)^(**)=k^(**)(A / x A)^{*}=k^{*}(A/xA)∗=k∗
The equivalence of (ii) and (iv) provides an answer to Question 4 for the special case of the polynomial XmY−F(X,Z,T)XmY−F(X,Z,T)X^(m)Y-F(X,Z,T)X^{m} Y-F(X, Z, T)XmY−F(X,Z,T). The equivalence of (i) and (iii) provides an answer
to a special case of Question 4′4′4^(')4^{\prime}4′ (stated below) for the ring R=k[x]R=k[x]R=k[x]R=k[x]R=k[x]. The equivalence of (iii) and (vi) answers Question 3 in a special situation. For more discussions, see [48].
In a remarkable paper Kaliman proved the following result over the field of complex numbers [56]. Later, Daigle and Kaliman extended it over any field kkkkk of characteristic zero [25].
Theorem 5.5. Let kkkkk be a field of characteristic zero. Let F∈k[X,Y,Z]F∈k[X,Y,Z]F in k[X,Y,Z]F \in k[X, Y, Z]F∈k[X,Y,Z] be such that k[X,Y,Z]/(F−λ)=k[2]k[X,Y,Z]/(F−λ)=k[2]k[X,Y,Z]//(F-lambda)=k^([2])k[X, Y, Z] /(F-\lambda)=k^{[2]}k[X,Y,Z]/(F−λ)=k[2] for almost every λ∈kλ∈klambda in k\lambda \in kλ∈k. Then k[X,Y,Z]=k[F][2]k[X,Y,Z]=k[F][2]k[X,Y,Z]=k[F]^([2])k[X, Y, Z]=k[F]^{[2]}k[X,Y,Z]=k[F][2].
A general version of Question 4 can be asked as:
Question 4′4′4^(')4^{\prime}4′. Let RRRRR be a ring and f∈A=R[n]f∈A=R[n]f in A=R^([n])f \in A=R^{[n]}f∈A=R[n] for some integer n≥2n≥2n >= 2n \geq 2n≥2. Suppose
Does this imply that A=R[f][n−1]A=R[f][n−1]A=R[f]^([n-1])A=R[f]^{[n-1]}A=R[f][n−1], i.e., is fffff a coordinate in AAAAA ?
There have been affirmative answers to Question 4′4′4^(')4^{\prime}4′ in special cases by Bhatwadekar, Dutta, and Das [11,13, 28]. Bhatwadekar and Dutta had considered linear planes, i.e., polynomials FFFFF of the form aZ−baZ−baZ-ba Z-baZ−b, where a,b∈R[X,Y]a,b∈R[X,Y]a,b in R[X,Y]a, b \in R[X, Y]a,b∈R[X,Y] over a discrete valuation ring RRRRR and proved that special cases of the linear planes are actually variables. Bhatwadekar-Dutta have also shown [12] that a negative answer to Question 4′4′4^(')4^{\prime}4′ in the case when n=3n=3n=3n=3n=3 and RRRRR is a discrete valuation ring containing QQQ\mathbb{Q}Q will give a negative answer to the affine fibration problem (Question 3 (i)) for the case n=2n=2n=2n=2n=2 and d=2d=2d=2d=2d=2. An example of a case of linear planes which remains unsolved is discussed in Section 7.
6. AnAnA^(n)\mathbb{A}^{\boldsymbol{n}}An-FORMS
Let AAAAA be an algebra over a field kkkkk. We say that AAAAA is an AnAnA^(n)\mathbb{A}^{n}An-form over kkkkk if A⊗kL=L[n]A⊗kL=L[n]Aox_(k)L=L^([n])A \otimes_{k} L=L^{[n]}A⊗kL=L[n] for some finite algebraic extension LLLLL of kkkkk. Let AAAAA be an AnAnA^(n)\mathbb{A}^{n}An-form over a field kkkkk.
When n=1n=1n=1n=1n=1, it is well known that if L|kLkL|_(k)\left.L\right|_{k}L|k is a separable extension, then A=k[1]A=k[1]A=k^([1])A=k^{[1]}A=k[1] (i.e., trivial) and that if L|kLkL|_(k)\left.L\right|_{k}L|k is purely inseparable then AAAAA need not be k[1]k[1]k^([1])k^{[1]}k[1]. An extensive study of such purely inseparable algebras was made by Asanuma in [8]. Over any field of positive characteristic, the nontrivial purely inseparable A1A1A^(1)\mathbb{A}^{1}A1-forms can be used to give examples of nontrivial AnAnA^(n)\mathbb{A}^{n}An-forms for any integer n>1n>1n > 1n>1n>1.
When n=2n=2n=2n=2n=2 and L|kLkL|_(k)\left.L\right|_{k}L|k is a separable extension, then Kambayashi established that A=k[2]A=k[2]A=k^([2])A=k^{[2]}A=k[2] [57]. However, the problem of existence of nontrivial separable A3A3A^(3)\mathbb{A}^{3}A3-forms is open in general. A few recent partial results on the triviality of separable A3A3A^(3)\mathbb{A}^{3}A3-forms are mentioned below.
Let AAAAA be an A3A3A^(3)\mathbb{A}^{3}A3-form over a field kkkkk of characteristic zero and k¯k¯bar(k)\bar{k}k¯ be an algebraic closure of kkkkk. Then A=k[3]A=k[3]A=k^([3])A=k^{[3]}A=k[3] if it satisfies any one of the following:
(1) AAAAA admits a fixed point free locally nilpotent derivation DDDDD (Daigle and Kaliman [25, COROLLARY 3.3]).
(2) AAAAA contains an element fffff which is a coordinate of A⊗kk¯A⊗kk¯Aox_(k) bar(k)A \otimes_{k} \bar{k}A⊗kk¯ (Daigle and Kaliman [25, PROPOSITION 4.9]).
(3) AAAAA admits an effective action of a reductive algebraic kkkkk-group of positive dimension (Koras and Russell [61, THEOREM c]).
(4) AAAAA admits either a fixed point free locally nilpotent derivation or a nonconfluent action of a unipotent group of dimension two (Gurjar, Masuda, and Miyanishi [51]).
(5) AAAAA admits a locally nilpotent derivation DDDDD such that rk(D⊗1k¯)≤2rkâ¡D⊗1k¯≤2rk(D ox1_( bar(k))) <= 2\operatorname{rk}\left(D \otimes 1_{\bar{k}}\right) \leq 2rkâ¡(D⊗1k¯)≤2 (Dutta, Gupta, and Lahiri [39]).
Now let RRRRR be a ring containing a field kkkkk. An RRRRR-algebra AAAAA is said to be an AnAnA^(n)\mathbb{A}^{n}An-form over RRRRR with respect to kkkkk if A⊗kk¯=(R⊗kk¯)[n]A⊗kk¯=R⊗kk¯[n]Aox_(k) bar(k)=(Rox_(k)( bar(k)))^([n])A \otimes_{k} \bar{k}=\left(R \otimes_{k} \bar{k}\right)^{[n]}A⊗kk¯=(R⊗kk¯)[n], where k¯k¯bar(k)\bar{k}k¯ denotes the algebraic closure of kkkkk. A few results on triviality of separable AnAnA^(n)A^{n}An-forms over a ring RRRRR are listed below.
Let AAAAA be an AnAnA^(n)\mathbb{A}^{n}An-form over a ring RRRRR containing a field kkkkk of characteristic 0 . Then:
(1) If n=1n=1n=1n=1n=1, then AAAAA is isomorphic to the symmetric algebra of a finitely generated rank one projective module over RRRRR [35, THEOREM 7].
(2) If n=2n=2n=2n=2n=2 and RRRRR is a PID containing QQQ\mathbb{Q}Q, then A=R[2]A=R[2]A=R^([2])A=R^{[2]}A=R[2] [35, REMARK 8].
(3) If n=2n=2n=2n=2n=2, then AAAAA is an A2A2A^(2)\mathbb{A}^{2}A2-fibration over RRRRR.
(4) If n=2n=2n=2n=2n=2 and RRRRR is a one-dimensional Noetherian domain, then there exists a finitely generated rank-one projective RRRRR-module QQQQQ such that A≅(SymR(Q))[1]A≅SymRâ¡(Q)[1]A~=(Sym_(R)(Q))^([1])A \cong\left(\operatorname{Sym}_{R}(Q)\right)^{[1]}A≅(SymRâ¡(Q))[1] [39, THEOREM 3.7].
(5) If n=2n=2n=2n=2n=2 and AAAAA admits has a fixed point free locally nilpotent RRRRR-derivation over any ring RRRRR, then there exists a finitely generated rank one projective RRRRR-module QQQQQ such that A≅(SymR(Q))[1]A≅SymRâ¡(Q)[1]A~=(Sym_(R)(Q))^([1])A \cong\left(\operatorname{Sym}_{R}(Q)\right)^{[1]}A≅(SymRâ¡(Q))[1] [39, THEOREM 3.8].
The result (3) above shows that an affirmative answer to the A2A2A^(2)\mathbb{A}^{2}A2-fibration problem (Question 3 (i)) will ensure an affirmative answer to the problem of A2A2A^(2)\mathbb{A}^{2}A2-forms over general rings. Over a field FFFFF of any characteristic, Das has shown [27] that any factorial A1A1A^(1)\mathbb{A}^{1}A1-form AAAAA over a ring RRRRR containing FFFFF is trivial if there exists a retraction map from AAAAA to RRRRR.
We cannot say much about A3A3A^(3)\mathbb{A}^{3}A3-forms over general rings till the time we solve it over fields.
7. AN EXAMPLE OF BHATWADEKAR AND DUTTA
The following example arose from the study of linear planes over a discrete valuation ring by Bhatwadekar and Dutta [12]. Question 5 stated below is an open problem for at least three decades. Let
A=C[T,X,Y,Z] and R=C[T,F]⊂AA=C[T,X,Y,Z] and R=C[T,F]⊂AA=C[T,X,Y,Z]quad" and "quad R=C[T,F]sub AA=\mathbb{C}[T, X, Y, Z] \quad \text { and } \quad R=\mathbb{C}[T, F] \subset AA=C[T,X,Y,Z] and R=C[T,F]⊂A
where F=TX2Z+X+T2Y+TXY2F=TX2Z+X+T2Y+TXY2F=TX^(2)Z+X+T^(2)Y+TXY^(2)F=T X^{2} Z+X+T^{2} Y+T X Y^{2}F=TX2Z+X+T2Y+TXY2.
Let
P:=XZ+Y2G:=TY+XPP:=XZ+Y2G:=TY+XP{:[P:=XZ+Y^(2)],[G:=TY+XP]:}\begin{aligned}
& P:=X Z+Y^{2} \\
& G:=T Y+X P
\end{aligned}P:=XZ+Y2G:=TY+XP
and
H:=T2Z−2TYP−XP2H:=T2Z−2TYP−XP2H:=T^(2)Z-2TYP-XP^(2)H:=T^{2} Z-2 T Y P-X P^{2}H:=T2Z−2TYP−XP2
and F=X+TGF=X+TGF=X+TGF=X+T GF=X+TG. Clearly, C[T,T−1][F,G,H]⊆C[T,T−1][X,Y,Z]CT,T−1[F,G,H]⊆CT,T−1[X,Y,Z]C[T,T^(-1)][F,G,H]subeC[T,T^(-1)][X,Y,Z]\mathbb{C}\left[T, T^{-1}\right][F, G, H] \subseteq \mathbb{C}\left[T, T^{-1}\right][X, Y, Z]C[T,T−1][F,G,H]⊆C[T,T−1][X,Y,Z].
Then the following statements hold:
(i) C[T,T−1][X,Y,Z]=C[T,T−1,F,G,H]=C[T,T−1][F][2]CT,T−1[X,Y,Z]=CT,T−1,F,G,H=CT,T−1[F][2]C[T,T^(-1)][X,Y,Z]=C[T,T^(-1),F,G,H]=C[T,T^(-1)][F]^([2])\mathbb{C}\left[T, T^{-1}\right][X, Y, Z]=\mathbb{C}\left[T, T^{-1}, F, G, H\right]=\mathbb{C}\left[T, T^{-1}\right][F]^{[2]}C[T,T−1][X,Y,Z]=C[T,T−1,F,G,H]=C[T,T−1][F][2].
(ii) C[T,X,Y,Z]C[T,X,Y,Z]C[T,X,Y,Z]\mathbb{C}[T, X, Y, Z]C[T,X,Y,Z] is an A2A2A^(2)\mathbb{A}^{2}A2-fibration over C[T,F]C[T,F]C[T,F]\mathbb{C}[T, F]C[T,F].
(iii) C[T,X,Y,Z][1]=C[T,F][3]C[T,X,Y,Z][1]=C[T,F][3]C[T,X,Y,Z]^([1])=C[T,F]^([3])\mathbb{C}[T, X, Y, Z]^{[1]}=\mathbb{C}[T, F]^{[3]}C[T,X,Y,Z][1]=C[T,F][3]
(iv) C[T,X,Y,Z]/(F)=C[T][2]=C[3]C[T,X,Y,Z]/(F)=C[T][2]=C[3]C[T,X,Y,Z]//(F)=C[T]^([2])=C^([3])\mathbb{C}[T, X, Y, Z] /(F)=\mathbb{C}[T]^{[2]}=\mathbb{C}^{[3]}C[T,X,Y,Z]/(F)=C[T][2]=C[3].
(v) C[T,X,Y,Z]/(F−f(T))=C[T][2]C[T,X,Y,Z]/(F−f(T))=C[T][2]C[T,X,Y,Z]//(F-f(T))=C[T]^([2])\mathbb{C}[T, X, Y, Z] /(F-f(T))=\mathbb{C}[T]^{[2]}C[T,X,Y,Z]/(F−f(T))=C[T][2] for every polynomial f(T)∈C[T]f(T)∈C[T]f(T)inC[T]f(T) \in \mathbb{C}[T]f(T)∈C[T].
(vii) For any u∈(T,F)R,A[1/u]=R[1/u][2]u∈(T,F)R,A[1/u]=R[1/u][2]u in(T,F)R,A[1//u]=R[1//u]^([2])u \in(T, F) R, A[1 / u]=R[1 / u]^{[2]}u∈(T,F)R,A[1/u]=R[1/u][2], i.e., C[T,X,Y,Z][1/u]=C[T,X,Y,Z][1/u]=C[T,X,Y,Z][1//u]=\mathbb{C}[T, X, Y, Z][1 / u]=C[T,X,Y,Z][1/u]=C[T,F,1/u][2]C[T,F,1/u][2]C[T,F,1//u]^([2])\mathbb{C}[T, F, 1 / u]^{[2]}C[T,F,1/u][2].
Question 5.
(a) Is A=C[T,F][2](=R[2])A=C[T,F][2]=R[2]A=C[T,F]^([2])(=R^([2]))A=\mathbb{C}[T, F]^{[2]}\left(=R^{[2]}\right)A=C[T,F][2](=R[2]) ?
(b) At least is A=C[F][3]A=C[F][3]A=C[F]^([3])A=\mathbb{C}[F]^{[3]}A=C[F][3] ?
If the answer is "No" to (a), then it is a counterexample to the following problems:
(1) A2A2A^(2)\mathbb{A}^{2}A2-fibration Problem over C[2]C[2]C^([2])\mathbb{C}^{[2]}C[2] by (ii).
(2) Cancellation Problem over C[2]C[2]C^([2])\mathbb{C}^{[2]}C[2] by (iii).
(3) Epimorphism problem over the ring C[T]C[T]C[T]\mathbb{C}[T]C[T] (see Question 4′4′4^(')4^{\prime}4′ ) by (iv).
If the answer is "No" to (b) and hence to (a), then it is a counterexample also to the Epimorphism Problem for C[4]→C[3]C[4]→C[3]C^([4])rarrC^([3])\mathbb{C}^{[4]} \rightarrow \mathbb{C}^{[3]}C[4]→C[3].
Z=(H+2TYP+XP2)/T2Z=H+2TYP+XP2/T2Z=(H+2TYP+XP^(2))//T^(2)Z=\left(H+2 T Y P+X P^{2}\right) / T^{2}Z=(H+2TYP+XP2)/T2
and hence equation (1) follows.
(ii) Clearly, AAAAA is a finitely generated RRRRR-algebra. It can be shown by standard arguments that AAAAA is a flat RRRRR-algebra [66, THEOREM 20.H]. We now show that A⊗Rk(p)=k(p)[2]A⊗Rk(p)=k(p)[2]Aox_(R)k(p)=k(p)^([2])A \otimes_{R} k(p)=k(p)^{[2]}A⊗Rk(p)=k(p)[2] for every prime ideal ppppp of RRRRR. We note that F−X∈TAF−X∈TAF-X in TAF-X \in T AF−X∈TA and hence the image of FFFFF in A/TAA/TAA//TAA / T AA/TA is same as that of XXXXX. Now let ppppp be a prime ideal of RRRRR. Then either T∈pT∈pT in pT \in pT∈p or T∉pT∉pT!in pT \notin pT∉p. If T∈pT∈pT in pT \in pT∈p, then A⊗Rk(p)=k(p)[Y,Z]=k(p)[2]A⊗Rk(p)=k(p)[Y,Z]=k(p)[2]Aox_(R)k(p)=k(p)[Y,Z]=k(p)^([2])A \otimes_{R} k(p)=k(p)[Y, Z]=k(p)^{[2]}A⊗Rk(p)=k(p)[Y,Z]=k(p)[2]. If T∉pT∉pT!in pT \notin pT∉p, then image of TTTTT in k(p)k(p)k(p)k(p)k(p) is a unit and the result follows from (i).
(iii) Let D=A[W]=C[T,X,Y,Z,W]=C[5]D=A[W]=C[T,X,Y,Z,W]=C[5]D=A[W]=C[T,X,Y,Z,W]=C^([5])D=A[W]=\mathbb{C}[T, X, Y, Z, W]=\mathbb{C}^{[5]}D=A[W]=C[T,X,Y,Z,W]=C[5]. We shall show that
and that C[T,F,G2,H1,W2]⊆DCT,F,G2,H1,W2⊆DC[T,F,G_(2),H_(1),W_(2)]sube D\mathbb{C}\left[T, F, G_{2}, H_{1}, W_{2}\right] \subseteq DC[T,F,G2,H1,W2]⊆D. Let D/TD=C[x,y,z,w]D/TD=C[x,y,z,w]D//TD=C[x,y,z,w]D / T D=\mathbb{C}[x, y, z, w]D/TD=C[x,y,z,w], where x,y,z,wx,y,z,wx,y,z,wx, y, z, wx,y,z,w denote the images of X,Y,Z,WX,Y,Z,WX,Y,Z,WX, Y, Z, WX,Y,Z,W in D/TDD/TDD//TDD / T DD/TD. We now show that D⊆C[T,F,G2,H1,W2]D⊆CT,F,G2,H1,W2D subeC[T,F,G_(2),H_(1),W_(2)]D \subseteq \mathbb{C}\left[T, F, G_{2}, H_{1}, W_{2}\right]D⊆C[T,F,G2,H1,W2]. For this, it is enough to show that the kernel of the natural map ϕ:C[T,F,G2,H1,W2]→D/TDÏ•:CT,F,G2,H1,W2→D/TDphi:C[T,F,G_(2),H_(1),W_(2)]rarr D//TD\phi: \mathbb{C}\left[T, F, G_{2}, H_{1}, W_{2}\right] \rightarrow D / T DÏ•:C[T,F,G2,H1,W2]→D/TD is generated by TTTTT. We note that the image of ϕÏ•phi\phiÏ• is
C[x,y−xw,z+2yw−xw2,w+2p(y−xw−xp2)(y−xw)−xp(z+2yw−xw2)]Cx,y−xw,z+2yw−xw2,w+2py−xw−xp2(y−xw)−xpz+2yw−xw2C[x,y-xw,z+2yw-xw^(2),w+2p(y-xw-xp^(2))(y-xw)-xp(z+2yw-xw^(2))]\mathbb{C}\left[x, y-x w, z+2 y w-x w^{2}, w+2 p\left(y-x w-x p^{2}\right)(y-x w)-x p\left(z+2 y w-x w^{2}\right)\right]C[x,y−xw,z+2yw−xw2,w+2p(y−xw−xp2)(y−xw)−xp(z+2yw−xw2)],
which is of transcendence degree 4 over CCC\mathbb{C}C. Hence the kernel of ϕÏ•phi\phiÏ• is a prime ideal of height one and is generated by TTTTT. Therefore, D=C[T,F,G2,H1,W2]D=CT,F,G2,H1,W2D=C[T,F,G_(2),H_(1),W_(2)]D=\mathbb{C}\left[T, F, G_{2}, H_{1}, W_{2}\right]D=C[T,F,G2,H1,W2].
(iv)-(v) Let B=C[T,X,Y,Z]/(F−f(T))B=C[T,X,Y,Z]/(F−f(T))B=C[T,X,Y,Z]//(F-f(T))B=\mathbb{C}[T, X, Y, Z] /(F-f(T))B=C[T,X,Y,Z]/(F−f(T)) for some polynomial f∈C[T]f∈C[T]f inC[T]f \in \mathbb{C}[T]f∈C[T] and S=C[T]S=C[T]S=C[T]S=\mathbb{C}[T]S=C[T]. By (ii), it follows that BBBBB is an A2A2A^(2)\mathbb{A}^{2}A2-fibration over SSSSS. Hence, by Sathaye's theorem [81], BBBBB is locally a polynomial ring over SSSSS and hence by Theorem 4.1, BBBBB is a polynomial ring over SSSSS.
(vi) Let H1:=FH+G2TH1:=FH+G2TH_(1):=(FH+G^(2))/(T)H_{1}:=\frac{F H+G^{2}}{T}H1:=FH+G2T. Then
H1=(X+TG)(T2Z−2TYP−XP2)+(TY+XP)2T=TP+GHH1=(X+TG)T2Z−2TYP−XP2+(TY+XP)2T=TP+GHH_(1)=((X+TG)(T^(2)Z-2TYP-XP^(2))+(TY+XP)^(2))/(T)=TP+GHH_{1}=\frac{(X+T G)\left(T^{2} Z-2 T Y P-X P^{2}\right)+(T Y+X P)^{2}}{T}=T P+G HH1=(X+TG)(T2Z−2TYP−XP2)+(TY+XP)2T=TP+GH
Let H2:=FH1+G3TH2:=FH1+G3TH_(2):=(FH_(1)+G^(3))/(T)H_{2}:=\frac{F H_{1}+G^{3}}{T}H2:=FH1+G3T. Then
H2=(X+TG)(TP+GH)+G3T=T(G2H+TGP+XP)+G(XH+G2)T=T(G2H+TGP+XP)+GT2PT=G2H+XP+2TGPH2=(X+TG)(TP+GH)+G3T=TG2H+TGP+XP+GXH+G2T=TG2H+TGP+XP+GT2PT=G2H+XP+2TGP{:[H_(2)=((X+TG)(TP+GH)+G^(3))/(T)],[=(T(G^(2)H+TGP+XP)+G(XH+G^(2)))/(T)],[=(T(G^(2)H+TGP+XP)+GT^(2)P)/(T)],[=G^(2)H+XP+2TGP]:}\begin{aligned}
H_{2} & =\frac{(X+T G)(T P+G H)+G^{3}}{T} \\
& =\frac{T\left(G^{2} H+T G P+X P\right)+G\left(X H+G^{2}\right)}{T} \\
& =\frac{T\left(G^{2} H+T G P+X P\right)+G T^{2} P}{T} \\
& =G^{2} H+X P+2 T G P
\end{aligned}H2=(X+TG)(TP+GH)+G3T=T(G2H+TGP+XP)+G(XH+G2)T=T(G2H+TGP+XP)+GT2PT=G2H+XP+2TGP
Let H3:=F(H2−G)+G4TH3:=FH2−G+G4TH_(3):=(F(H_(2)-G)+G^(4))/(T)H_{3}:=\frac{F\left(H_{2}-G\right)+G^{4}}{T}H3:=F(H2−G)+G4T. Then
H3=F(G2H+XP+2TGP−XP−TY)+G4T=F(2TGP−TY)+G2(FH+G2)T=TF(2GP−Y)+TH1G2T=F(2GP−Y)+H1G2H3=FG2H+XP+2TGP−XP−TY+G4T=F(2TGP−TY)+G2FH+G2T=TF(2GP−Y)+TH1G2T=F(2GP−Y)+H1G2{:[H_(3)=(F(G^(2)H+XP+2TGP-XP-TY)+G^(4))/(T)],[=(F(2TGP-TY)+G^(2)(FH+G^(2)))/(T)],[=(TF(2GP-Y)+TH_(1)G^(2))/(T)],[=F(2GP-Y)+H_(1)G^(2)]:}\begin{aligned}
H_{3} & =\frac{F\left(G^{2} H+X P+2 T G P-X P-T Y\right)+G^{4}}{T} \\
& =\frac{F(2 T G P-T Y)+G^{2}\left(F H+G^{2}\right)}{T} \\
& =\frac{T F(2 G P-Y)+T H_{1} G^{2}}{T} \\
& =F(2 G P-Y)+H_{1} G^{2}
\end{aligned}H3=F(G2H+XP+2TGP−XP−TY)+G4T=F(2TGP−TY)+G2(FH+G2)T=TF(2GP−Y)+TH1G2T=F(2GP−Y)+H1G2
and that the image of C[T,F,F−1,G,H2]CT,F,F−1,G,H2C[T,F,F^(-1),G,H_(2)]\mathbb{C}\left[T, F, F^{-1}, G, H_{2}\right]C[T,F,F−1,G,H2] in A[F−1]/TA[F−1]AF−1/TAF−1A[F^(-1)]//TA[F^(-1)]A\left[F^{-1}\right] / T A\left[F^{-1}\right]A[F−1]/TA[F−1] is of transcendence degree 3 . Hence A[F−1]=C[T,F,F−1,G,H3]=C[T,F,F−1,G][1]AF−1=CT,F,F−1,G,H3=CT,F,F−1,G[1]A[F^(-1)]=C[T,F,F^(-1),G,H_(3)]=C[T,F,F^(-1),G]^([1])A\left[F^{-1}\right]=\mathbb{C}\left[T, F, F^{-1}, G, H_{3}\right]=\mathbb{C}\left[T, F, F^{-1}, G\right]^{[1]}A[F−1]=C[T,F,F−1,G,H3]=C[T,F,F−1,G][1].
(vii) Let mmmmm be any maximal ideal of RRRRR other than (T,F)(T,F)(T,F)(T, F)(T,F). Then either T∉mT∉mT!in mT \notin mT∉m or F∉mF∉mF!in mF \notin mF∉m. Thus, in either case, from (i) and (vi), we have Am=Rm[2]Am=Rm[2]A_(m)=R_(m)^([2])A_{m}=R_{m}^{[2]}Am=Rm[2].
Let u∈(T,F)Ru∈(T,F)Ru in(T,F)Ru \in(T, F) Ru∈(T,F)R. Then a maximal ideal of R[1/u]R[1/u]R[1//u]R[1 / u]R[1/u] is an extension of a maximal ideal of RRRRR other than (T,F)R(T,F)R(T,F)R(T, F) R(T,F)R. Hence A[1/u]A[1/u]A[1//u]A[1 / u]A[1/u] is a locally polynomial ring in two variables over R[1/u]R[1/u]R[1//u]R[1 / u]R[1/u]. Further any projective module over R[1/u]R[1/u]R[1//u]R[1 / u]R[1/u] is free. Thus, by Theorem 4.1 , we have A[1/u]=R[1/u][2]A[1/u]=R[1/u][2]A[1//u]=R[1//u]^([2])A[1 / u]=R[1 / u]^{[2]}A[1/u]=R[1/u][2].
ACKNOWLEDGMENTS
The author thanks Professor Amartya Kumar Dutta for introducing and guiding her to this world of affine algebraic geometry. The author also thanks him for carefully going through this draft and improving the exposition.
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NEENA GUPTA
Statistics and Mathematics Unit, Indian Statistical Institute, 203 B.T. Road,
The formal model of semi-infinite flag manifold is a variant of an affine flag variety that was recognized from the 1980s but not studied extensively until the late 2010s. In this note, we exhibit constructions and ideas appearing in our recent study of the formal model of semi-infinite flag manifold of a simple algebraic group. Our results have some implications to the theory of rational maps from a projective line to partial flag manifolds, and also on the structures of quantum cohomologies and quantum KKKKK-groups of partial flag manifolds.
Semi-infinite flag manifold, quasi-map space, quantum K-group, Kac-Moody group, affine
Lie algebra, global Weyl module
1. INTRODUCTION
Compact complex-analytic spaces that admit homogeneous Lie group actions are quite rare in nature, and their classification reduces into three primitive classes: finite groups, tori, and (partial) flag manifolds. The first have discrete topology and the role of geometric consideration is rather small, in general. The second, particularly those admit polarizations, offer a major subject known as abelian varieties. The third, the (partial) flag manifolds of compact simple Lie groups, are ubiquitous in representation theory of semisimple algebraic groups and quantum groups. By the universal nature of general linear groups, flag manifolds of unitary groups are extensively studied from the geometric perspective.
In representation-theoretic considerations, we usually consider flag manifolds as projective algebraic varieties defined over an algebraically closed field (that form a family over SpecZSpecâ¡ZSpec Z\operatorname{Spec} \mathbb{Z}Specâ¡Z ). This definition naturally extends to an arbitrary Kac-Moody setting, but the resulting objects have at least two variants, thin flag varieties and thick flag manifolds (defined by Kac-Peterson [75] and Kashiwara [40], respectively). In case the Kac-Moody group is of affine type, we have a loop realization of the corresponding Kac-Moody group, essentially identifying the corresponding group with the set of k((z))k((z))k((z))\mathbb{k}((z))k((z))-valued points of a simple algebraic group over a field kkk\mathbb{k}k. This motivates us to consider yet other versions of flag manifolds of affine type that can be understood as an enhancement of arc schemes of usual flag manifolds. These are the semi-infinite flag manifolds that originate from the ideas of Lusztig [63, $11] and Drinfeld [22]. Lusztig's original idea is to construct varieties that naturally encode representation theory of simple algebraic groups over finite fields. The Lusztig program (see, e.g., [44,63][44,63][44,63][44,63][44,63] ) adds representation theory of quantum groups at roots of unity and representation theory of affine Lie algebras at negative rational levels into the picture, and Feigin-Frenkel [19] put representation theory of affine Lie algebras at the critical level into the picture. The semi-infinite flag manifolds itself have two realizations, that we refer to as the ind-model and the formal model. The geometry of the ind-model of semi-infinite flag manifolds, also known as the space of quasimaps from a projective line to a flag manifold, was studied extensively by Braverman, Finkelberg, Mirković, and their collaborators (see [8,18,21,22][8,18,21,22][8,18,21,22][8,18,21,22][8,18,21,22] ).
One instance of the ind-model of semi-infinite flag manifold is the space of principal bundles on an algebraic curve equipped with some reduction. This interpretation realizes some portion of the above representation-theoretic expectations [2,31]. The formal model of semi-infinite flag manifolds is expected to add a concrete understanding of related representation-theoretic patterns [19,22,63][19,22,63][19,22,63][19,22,63][19,22,63]. Unfortunately, such an idea needs to be polished as its implementation faces difficulty due to its essential infinite-dimensionality. This forces us to employ affine Grassmannians instead of semi-infinite flag manifolds in some cases (see [26,30,78][26,30,78][26,30,78][26,30,78][26,30,78] ) at the moment, that is possible by some tight connections [27,70][27,70][27,70][27,70][27,70].
Meanwhile, it is realized that the semi-infinite flag manifold is a version of the loop space of a flag manifold, and hence it is related to its quantum cohomology [32]. In fact, the ind-model of a semi-infinite flag manifold offers a description of the quantum KKKKK-theoretic JJJJJ-function of a flag manifold [9] that encodes its small quantum KKKKK-group.
In both contexts of the above two paragraphs, the Peterson isomorphism [59, 74], that connects the quantum cohomology of a flag manifold with the homology of an affine Grassmannian, should admit an interpretation using a semi-infinite flag manifold. However, such an interpretation is not known today (though we have Corollary 7.3).
The main goal of this note is to explain a realization of the formal model of semiinfinite flag manifold [46,50,52], that is reminiscent to the classical description of the original flag manifolds. Our realization is supported by recent developments in representation theory of affine Lie algebras [14,15,51][14,15,51][14,15,51][14,15,51][14,15,51], that is also reminiscent to the representation theory of simple Lie algebras. It turns out that the study of the formal model of the semi-infinite flag manifold has implications to the corresponding ind-model [50], as well as the study of quantum KKKKK-groups of partial flag manifolds and the KKKKK-groups of affine Grassmannians [45, 47, 48]. This includes an interpretation (and a proof) of the KKKKK-theoretic analogue of the Peterson isomorphism using semi-infinite flag manifolds (Theorem 8.2).
The results presented here describe the formal model of semi-infinite flag manifolds in a down-to-earth fashion, and also provide first nontrivial conclusions deduced from them. However, we have not yet reached our primary goal to understand representation theory from this perspective in a satisfactory fashion. We hope to improve this situation in the near future.
The organization of this note is as follows: We first recall the construction of flag manifolds that is parallel to our later construction in Section 2. We explain the role of quantum groups in the structure theory of Kac-Moody algebras and exhibit two versions of flag varieties of Kac-Moody groups in Section 3. In Section 4, we exhibit some representation theory of affine Lie algebras. Based on it, we explain our construction of the formal model of semi-infinite flag manifolds in Section 5. This enables us to present our idea on the Frobenius splitting of semi-infinite flag manifolds in Section 6. We explain the connection between its Richardson varieties and quasimap spaces in Section 7, and explain how they fit into the study of quantum cohomology of flag manifolds. We exhibit the KKKKK-theoretic Peterson isomorphism in Section 8. We discuss the functoriality of the quantum KKKKK-groups of partial flag manifolds in Section 9. We finish this note by discussing some perspectives in Section 10.
We assume that every field kkk\mathbb{k}k has characteristic ≠2≠2!=2\neq 2≠2. A variety is some algebraicgeometric object that admits singularity, and a manifold is a variety that is supposed to be smooth in some sense. An algebraic variety is a separated scheme of finite type defined over a field (i.e., our variety is not necessarily irreducible or reduced). We set N:=Z≥0N:=Z≥0N:=Z_( >= 0)\mathbb{N}:=\mathbb{Z}_{\geq 0}N:=Z≥0.
2. FLAG MANIFOLDS VIA REPRESENTATION THEORY
Let GGGGG be a simply connected semisimple algebraic group over an algebraically closed field kkk\mathbb{k}k. Let T⊂BT⊂BT sub BT \subset BT⊂B be its maximal torus and a Borel subgroup (maximal solvable subgroup). Let W(=NG(T)/T)W=NG(T)/TW(=N_(G)(T)//T)W\left(=N_{G}(T) / T\right)W(=NG(T)/T) be the Weyl group of GGGGG. Let XXX\mathbb{X}X be the set of onedimensional rational TTTTT-characters (the set of TTTTT-weights), that admits a natural WWWWW-action. We set X+:=∑i=1rNiX+:=∑i=1r NiX_(+):=sum_(i=1)^(r)N_(i)\mathbb{X}_{+}:=\sum_{i=1}^{r} \mathbb{N}_{i}X+:=∑i=1rNi, where ϖ1,…,ϖr∈XÏ–1,…,Ï–r∈XÏ–_(1),dots,Ï–_(r)inX\varpi_{1}, \ldots, \varpi_{r} \in \mathbb{X}Ï–1,…,Ï–r∈X are fundamental weights with respect to BBBBB. The set of isomorphism classes of irreducible rational representations {L(λ)}λ{L(λ)}λ{L(lambda)}_(lambda)\{L(\lambda)\}_{\lambda}{L(λ)}λ of GGGGG is labeled by X+X+X_(+)\mathbb{X}_{+}X+in such a way that each L(λ)L(λ)L(lambda)L(\lambda)L(λ) contains a unique (up to scalar) BBBBB-eigenvector vλvλv_(lambda)\mathbf{v}_{\lambda}vλ with its TTTTT-weight λλlambda\lambdaλ. We refer λ(∈X+)λ∈X+lambda(inX_(+))\lambda\left(\in \mathbb{X}_{+}\right)λ(∈X+)as the highest weight of L(λ)L(λ)L(lambda)L(\lambda)L(λ). The flag manifold B:=G/BB:=G/BB:=G//B\mathscr{B}:=G / BB:=G/B of GGGGG is the maximal GGGGG-homogeneous space that is projective.
In case k=Ck=Ck=C\mathbb{k}=\mathbb{C}k=C, we have
where yyyyy is an affine algebraic variety with (G×T)(G×T)(G xx T)(G \times T)(G×T)-action whose ring C[y]C[y]C[y]\mathbb{C}[y]C[y] of regular functions is written as
(2.1)C[y]≅λ∈X+L(λ)∗( as G×T-modules )(2.1)C[y]≅λ∈X+L(λ)∗( as G×T-modules ){:(2.1)C[y]~=_(lambda inX_(+))L(lambda)^(**)quad(" as "G xx T"-modules "):}\begin{equation*}
\mathbb{C}[y] \cong \underset{\lambda \in \mathbb{X}_{+}}{ } L(\lambda)^{*} \quad(\text { as } G \times T \text {-modules }) \tag{2.1}
\end{equation*}(2.1)C[y]≅λ∈X+L(λ)∗( as G×T-modules )
and E⊂yE⊂yE sub yE \subset yE⊂y is the locus where the TTTTT-action is not free. Here, the GGGGG-action on C[y]C[y]C[y]\mathbb{C}[y]C[y] is the natural actions on L(λ)L(λ)L(lambda)L(\lambda)L(λ), and the TTTTT-action on C[y]C[y]C[y]\mathbb{C}[y]C[y] comes from the grading X+⊂XX+⊂XX_(+)subX\mathbb{X}_{+} \subset \mathbb{X}X+⊂X in the RHS of (2.1). These data, together with the condition E≠yE≠yE!=yE \neq yE≠y, essentially determine C[y]C[y]C[y]\mathbb{C}[y]C[y] as CCC\mathbb{C}C-algebras generated by L(ϖi)∗LÏ–i∗L(Ï–_(i))^(**)L\left(\varpi_{i}\right)^{*}L(Ï–i)∗ for 1≤i≤r1≤i≤r1 <= i <= r1 \leq i \leq r1≤i≤r. Consider a point x0∈yx0∈yx_(0)in yx_{0} \in yx0∈y given by {vλ}λvλλ{v_(lambda)}lambda\left\{\mathbf{v}_{\lambda}\right\} \lambda{vλ}λ, seen as linear maps on {L(λ)∗}λL(λ)∗λ{L(lambda)^(**)}_(lambda)\left\{L(\lambda)^{*}\right\}_{\lambda}{L(λ)∗}λ. The image [x0]x0[x_(0)]\left[x_{0}\right][x0] of this point x0x0x_(0)x_{0}x0 has its GGGGG-stabilizer equal to BBBBB. This induces an inclusion
G/B↪B⊂∏i=1rP(L(ϖi))G/B↪B⊂âˆi=1r PLÏ–iG//B↪Bsubprod_(i=1)^(r)P(L(Ï–_(i)))G / B \hookrightarrow \mathscr{B} \subset \prod_{i=1}^{r} \mathbb{P}\left(L\left(\varpi_{i}\right)\right)G/B↪B⊂âˆi=1rP(L(Ï–i))
induced from B/B↦[x0]B/B↦x0B//B|->[x_(0)]B / B \mapsto\left[x_{0}\right]B/B↦[x0] by the GGGGG-action. (One needs additional representation-theoretic analysis to conclude G/B≅BG/B≅BG//B~=BG / B \cong \mathscr{B}G/B≅B.) This consideration transfers all geometric statements relevant to BBB\mathscr{B}B to algebraic statements on the space in (2.1) in principle, but most of the geometric results on BBB\mathscr{B}B and its subvarieties were proved for the first time by other methods (see, e.g., [56]).
Note that the vector space (2.1) does not acquire the structure of a ring when char k=k=k=\mathbb{k}=k=p>0p>0p > 0p>0p>0. The reason is that we do not have a map L(λ)∗⊗L(μ)∗→L(λ+μ)∗L(λ)∗⊗L(μ)∗→L(λ+μ)∗L(lambda)^(**)ox L(mu)^(**)rarr L(lambda+mu)^(**)L(\lambda)^{*} \otimes L(\mu)^{*} \rightarrow L(\lambda+\mu)^{*}L(λ)∗⊗L(μ)∗→L(λ+μ)∗, or equivalently, L(λ+μ)→L(λ)⊗L(μ)L(λ+μ)→L(λ)⊗L(μ)L(lambda+mu)rarr L(lambda)ox L(mu)L(\lambda+\mu) \rightarrow L(\lambda) \otimes L(\mu)L(λ+μ)→L(λ)⊗L(μ) for general λ,μ∈X+λ,μ∈X+lambda,mu inX_(+)\lambda, \mu \in \mathbb{X}_{+}λ,μ∈X+. One way to improve the situation is to replace {L(λ)}λ∈X+{L(λ)}λ∈X+{L(lambda)}_(lambda inX_(+))\{L(\lambda)\}_{\lambda \in \mathbb{X}_{+}}{L(λ)}λ∈X+with a suitable family of modules {Y(λ)}λ∈X+{Y(λ)}λ∈X+{Y(lambda)}_(lambda inX_(+))\{Y(\lambda)\}_{\lambda \in \mathbb{X}_{+}}{Y(λ)}λ∈X+with larger members such that the GGGGG-module map
exists uniquely (up to constant) for every λ,μ∈X+λ,μ∈X+lambda,mu inX_(+)\lambda, \mu \in \mathbb{X}_{+}λ,μ∈X+. It yields an analogous ring of (2.1) that should be closely related to BBB\mathscr{B}B. A standard choice of Y(λ)(λ∈X+)Y(λ)λ∈X+Y(lambda)(lambda inX_(+))Y(\lambda)\left(\lambda \in \mathbb{X}_{+}\right)Y(λ)(λ∈X+)is the Weyl module V(λ)V(λ)V(lambda)V(\lambda)V(λ) of GGGGG, that is, the projective cover of L(λ)L(λ)L(lambda)L(\lambda)L(λ) in the categories of rational GGGGG-modules whose composition factors are in {L(μ)}λ≥μ∈X+{L(μ)}λ≥μ∈X+{L(mu)}_(lambda >= mu inX_(+))\{L(\mu)\}_{\lambda \geq \mu \in \mathbb{X}_{+}}{L(μ)}λ≥μ∈X+, where ≥≥>=\geq≥is the dominance ordering on XXX\mathbb{X}X. This produces BBB\mathscr{B}B for all characteristics.
Theorem 2.1 (Orthogonality of Weyl modules, [36, II $4.13]). For each λ,μ∈X+λ,μ∈X+lambda,mu inX_(+)\lambda, \mu \in \mathbb{X}_{+}λ,μ∈X+, we have
Note that L(λ)=V(λ)L(λ)=V(λ)L(lambda)=V(lambda)L(\lambda)=V(\lambda)L(λ)=V(λ) for char k=0k=0k=0\mathbb{k}=0k=0 by the semisimplicity of representations, and hence Theorem 2.1 is Schur's lemma in such a case. As V(λ)=k⊗ZVZ(λ)V(λ)=k⊗ZVZ(λ)V(lambda)=kox_(Z)V_(Z)(lambda)V(\lambda)=\mathbb{k} \otimes_{\mathbb{Z}} V_{\mathbb{Z}}(\lambda)V(λ)=k⊗ZVZ(λ) holds for a collection of free ZZZ\mathbb{Z}Z-modules VZ(λ)(λ∈X+)VZ(λ)λ∈X+V_(Z)(lambda)(lambda inX_(+))V_{\mathbb{Z}}(\lambda)\left(\lambda \in \mathbb{X}_{+}\right)VZ(λ)(λ∈X+), we find that BBB\mathscr{B}B extends to a scheme flat over ZZZ\mathbb{Z}Z. Another possible choice of Y(λ)(λ∈X+Y(λ)λ∈X+Y(lambda)(lambda inX_(+):}Y(\lambda)\left(\lambda \in \mathbb{X}_{+}\right.Y(λ)(λ∈X+), the Verma module M(λ)M(λ)M(lambda)M(\lambda)M(λ) of the (divided power) enveloping algebra of Lie GGGGG, produces an open dense BBBBB-orbit in BBB\mathscr{B}B.
3. KAC-MOODY FLAG VARIETIES
Let us keep the setting of the previous section.
3.1. Reminder on Kac-Moody algebras and their quantum groups
Let gCgCg_(C)g_{C}gC be the Kac-Moody algebra associated to a symmetrizable generalized Cartan matrix ( ===== GCM) CCCCC (see [38]). In case char k=0k=0k=0k=0k=0, we have the notion of the highest weight integrable representations of gCgCg_(C)g_{C}gC parametrized by the set of dominant weights P+P+P_(+)P_{+}P+ defined similarly to X+X+X_(+)\mathbb{X}_{+}X+. Let L(Λ)L(Λ)L(Lambda)L(\Lambda)L(Λ) denote the highest weight integrable representation of gCgCg_(C)\mathrm{g}_{C}gC corresponding to Λ∈P+Λ∈P+Lambda inP_(+)\Lambda \in P_{+}Λ∈P+.
We have the quantum group (or the quantized enveloping algebra) Uq(gC)UqgCU_(q)(g_(C))U_{q}\left(g_{C}\right)Uq(gC) of gCgCg_(C)g_{C}gC originally defined by Drinfeld and Jimbo in the 1980s [17, 37]. It is an algebra defined over Q(q)Q(q)Q(q)\mathbb{Q}(q)Q(q), and the specialization q↦1q↦1q|->1q \mapsto 1q↦1 recovers the universal enveloping algebra U(gC)UgCU(g_(C))U\left(\mathrm{~g}_{C}\right)U( gC) of gCgCg_(C)\mathrm{g}_{C}gC. Kashiwara [41] and Lusztig [63] defined the canonical/global bases (of the positive/negative parts Uq±(gC))Uq±gC{:U_(q)^(+-)(g_(C)))\left.U_{q}^{ \pm}\left(g_{C}\right)\right)Uq±(gC)) of Uq(gC)UqgCU_(q)(g_(C))U_{q}\left(g_{C}\right)Uq(gC) and their integrable representations that generate their Q[q]−Q[q]−Q[q]-\mathbb{Q}[q]-Q[q]− lattices. The construction of Lusztig [64] clarified that quantum groups are, in fact, defined over Z[q±1]Zq±1Z[q^(+-1)]\mathbb{Z}\left[q^{ \pm 1}\right]Z[q±1] (or even over N[q±1]Nq±1N[q^(+-1)]\mathbb{N}\left[q^{ \pm 1}\right]N[q±1] if one can say). In the 2010s, the categorification theorems of a quantum group and its integrable representations appeared [39,53,76,77][39,53,76,77][39,53,76,77][39,53,76,77][39,53,76,77], and there every algebra that admits a categorification has a suitable Z[q]Z[q]Z[q]\mathbb{Z}[q]Z[q]-integral structure with distinguished bases, being the Grothendieck group of a module category of a finitely-generated graded algebras (called KLR algebras or quiver Hecke algebras). Therefore, the following is now widely recognized:
Theorem 3.1 (Lusztig [63,64,66][63,64,66][63,64,66][63,64,66][63,64,66] and Kashiwara [41-43]). Assume that k=Ck=Ck=C\mathbb{k}=\mathbb{C}k=C. The (lower) global bases of Uq±(gC)Uq±gCU_(q)^(+-)(g_(C))U_{q}^{ \pm}\left(g_{C}\right)Uq±(gC) induce a ZZZ\mathbb{Z}Z-integral form UZ(gC)UZgCU_(Z)(g_(C))U_{\mathbb{Z}}\left(g_{C}\right)UZ(gC) of U(gC)UgCU(g_(C))U\left(g_{C}\right)U(gC) via q↦1q↦1q|->1q \mapsto 1q↦1. For each Λ∈P+Λ∈P+Lambda inP_(+)\Lambda \in P_{+}Λ∈P+, we have a ZZZ\mathbb{Z}Z-lattice L(Λ)ZL(Λ)ZL(Lambda)_(Z)L(\Lambda)_{\mathbb{Z}}L(Λ)Z of L(Λ)L(Λ)L(Lambda)L(\Lambda)L(Λ) obtained from the (lower) global base of the corresponding integrable highest weight module of Uq(gC)UqgCU_(q)(g_(C))U_{q}\left(g_{C}\right)Uq(gC). In addition, L(Λ)ZL(Λ)ZL(Lambda)_(Z)L(\Lambda)_{\mathbb{Z}}L(Λ)Z is generated by the UZ(gC)UZgCU_(Z)(g_(C))U_{\mathbb{Z}}\left(\mathrm{g}_{C}\right)UZ(gC)-action from a highest weight vector of L(Λ)L(Λ)L(Lambda)L(\Lambda)L(Λ).
By a specialization of L(Λ)ZL(Λ)ZL(Lambda)_(Z)L(\Lambda)_{\mathbb{Z}}L(Λ)Z, we obtain a highest weight integrable module L(Λ)L(Λ)L(Lambda)L(\Lambda)L(Λ) over an arbitrary field kkk\mathbb{k}k. The module L(Λ)L(Λ)L(Lambda)L(\Lambda)L(Λ) is no longer irreducible when char k>0k>0k > 0\mathbb{k}>0k>0 (in general), and hence it is a gCgCg_(C)\mathrm{g}_{C}gC-analogue of Weyl modules rather than L(λ)L(λ)L(lambda)L(\lambda)L(λ) for GGGGG; it is a lack of brevity of the author to choose this notation here. We close this subsection by noting that the integral forms at the end of Section 2 coincide with the integral forms in Theorem 3.1.
3.2. Thin and thick flag varieties
Presentations of the flag varieties for general Kac-Moody groups EEE\mathscr{E}E associated to a GCM CCCCC are similar to those in the previous section. A triangular decomposition of gCgCg_(C)g_{C}gC yields an analogous group ℓâ„“â„“\ellâ„“ to the Borel subgroup. Let TTT\mathcal{T}T be a (standard) maximal torus of ℓâ„“â„“\ellâ„“. The highest weight vector in L(Λ)L(Λ)L(Lambda)L(\Lambda)L(Λ) is precisely an ℓâ„“â„“\ellâ„“-eigenvector with its TTT\mathcal{T}T-weight ΛΛLambda\LambdaΛ. Therefore, the construction in the previous section produces E/ℓE/â„“E//â„“\mathscr{E} / \mathscr{\ell}E/â„“ via the ring
where L(Λ)∗L(Λ)∗L(Lambda)^(**)L(\Lambda)^{*}L(Λ)∗ is the vector space dual of L(Λ)L(Λ)L(Lambda)L(\Lambda)L(Λ), and L(Λ)∨L(Λ)∨L(Lambda)^(vv)L(\Lambda)^{\vee}L(Λ)∨ is the restricted dual of L(Λ)L(Λ)L(Lambda)L(\Lambda)L(Λ), defined to be the direct sum of (finite-dimensional) vector space duals offered by the TTT\mathcal{T}T-weight decomposition of L(Λ)L(Λ)L(Lambda)L(\Lambda)L(Λ).
In this case, both vector spaces in (3.1) are naturally rings. This corresponds to the choice of EEE\mathscr{E}E. The former ring defines BCthick =E/d[40,49,71]BCthick =E/d[40,49,71]B_(C)^("thick ")=E//d[40,49,71]\mathscr{B}_{C}^{\text {thick }}=\mathscr{E} / \mathscr{d}[40,49,71]BCthick =E/d[40,49,71] if we take EEE\mathscr{E}E to be a version of the Kac-Moody group that is completed with respect to the opposite direction to ℓâ„“â„“\mathscr{\ell}â„“. (This is the maximal Kac-Moody group, but the completion is taken in the opposite way as in the literature.) The latter ring can be seen as the projective limit of finitely-generated algebras, and the union of the spectrums of these rings yields BCthin =E/LBCthin =E/LB_(C)^("thin ")=E//L\mathscr{B}_{C}^{\text {thin }}=\mathscr{E} / \mathscr{L}BCthin =E/L [56,75] if we take EEE\mathscr{E}E as the uncompleted Kac-Moody group (the Kac-Peterson group or the minimal Kac-Moody group), or as the maximal Kac-Moody group completed with respect to the direction of ℓâ„“â„“\ellâ„“. In other words, we have variants of flag manifolds of Kac-Moody groups associated to a GCM CCCCC as:
The scheme BCthick BCthick B_(C)^("thick ")\mathscr{B}_{C}^{\text {thick }}BCthick is a union of infinite-dimensional affine spaces, and hence is smooth. However, BCthick BCthick B_(C)^("thick ")\mathscr{B}_{C}^{\text {thick }}BCthick is not compact in an essential way [24]. This picture is compatible with the fact that the Kac-Peterson group is defined by one-parameter generators (and relations), and hence BCthin BCthin B_(C)^("thin ")\mathscr{B}_{C}^{\text {thin }}BCthin is a union of finite-dimensional subvarieties BC,nthin BC,nthin B_(C,n)^("thin ")\mathscr{B}_{C, n}^{\text {thin }}BC,nthin consisting of points presented by a product of at most nnnnn generating elements. As such, each scheme BC,nthin BC,nthin B_(C,n)^("thin ")\mathscr{B}_{C, n}^{\text {thin }}BC,nthin is singular, and hence BCthin BCthin B_(C)^("thin ")\mathscr{B}_{C}^{\text {thin }}BCthin is understood to be singular. In fact, it does not admit an inductive limit description by finite-dimensional smooth pieces [24].
4. GLOBAL WEYL MODULES AND THEIR PROJECTIVITY
Let us consider the untwisted affine Kac-Moody case hereafter, with the same conventions as in the previous sections. In particular, our Kac-Moody groups are extensions of the groups
G((z)):=G(k((z))) and G[z±1]:=G(k[z±1])G((z)):=G(k((z))) and Gz±1:=Gkz±1G((z)):=G(k((z)))quad" and "quad G[z^(+-1)]:=G(k[z^(+-1)])G((z)):=G(\mathbb{k}((z))) \quad \text { and } \quad G\left[z^{ \pm 1}\right]:=G\left(\mathbb{k}\left[z^{ \pm 1}\right]\right)G((z)):=G(k((z))) and G[z±1]:=G(k[z±1])
by the loop rotation GmGmG_(m)\mathbb{G}_{m}Gm-actions (that we denote by Gmrot Gmrot G_(m)^("rot ")\mathbb{G}_{m}^{\text {rot }}Gmrot ) and the central extension GmGmG_(m)\mathbb{G}_{m}Gm actions. (These correspond to the maximal/minimal realizations of the Kac-Moody groups
in the previous section.) These are not (pro-)algebraic groups, and it sometimes causes difficulty. Nevertheless, each rational representation VVVVV of GGGGG induces representations
V((z)):=V⊗kk((z)) and V[z±1]:=V⊗kk[z±1]V((z)):=V⊗kk((z)) and Vz±1:=V⊗kkz±1V((z)):=Vox_(k)k((z))quad" and "quad V[z^(+-1)]:=Vox_(k)k[z^(+-1)]V((z)):=V \otimes_{\mathbb{k}} \mathbb{k}((z)) \quad \text { and } \quad V\left[z^{ \pm 1}\right]:=V \otimes_{\mathfrak{k}} \mathbb{k}\left[z^{ \pm 1}\right]V((z)):=V⊗kk((z)) and V[z±1]:=V⊗kk[z±1]
of G((z))G((z))G((z))G((z))G((z)) and G[z±1]Gz±1G[z^(+-1)]G\left[z^{ \pm 1}\right]G[z±1], respectively. These representations are not of highest weight, but still integrable representations when we lift them to the central extensions of G((z))G((z))G((z))G((z))G((z)) and G[z±1]Gz±1G[z^(+-1)]G\left[z^{ \pm 1}\right]G[z±1] by letting the center GmGmG_(m)\mathbb{G}_{m}Gm act trivially (i.e., they are level-zero integrable representations viewed as representations of affine Lie algebras).
In addition to the TTTTT-action, the representation V[z±1]Vz±1V[z^(+-1)]V\left[z^{ \pm 1}\right]V[z±1] carries Gmrot Gmrot G_(m)^("rot ")\mathbb{G}_{m}^{\text {rot }}Gmrot -action. Let δδdelta\deltaδ be the degree-one character of Gmrot Gmrot G_(m)^("rot ")\mathbb{G}_{m}^{\text {rot }}Gmrot , and set q:=eδq:=eδq:=e^(delta)q:=e^{\delta}q:=eδ. By abuse of notation, we might consider qnqnq^(n)q^{n}qn(n∈Z)(n∈Z)(n inZ)(n \in \mathbb{Z})(n∈Z) as the functor that twists the Gmrot Gmrot G_(m)^("rot ")\mathbb{G}_{m}^{\text {rot }}Gmrot -action by degree nnnnn. We define a graded character of a semisimple (T×Gmrot )T×Gmrot (T xxG_(m)^("rot "))\left(T \times \mathbb{G}_{m}^{\text {rot }}\right)(T×Gmrot )-module UUUUU as
Then, gch V[z±1]Vz±1V[z^(+-1)]V\left[z^{ \pm 1}\right]V[z±1] makes sense as all the coefficients are in ZZZ\mathbb{Z}Z. However, if we take the second symmetric power S2(V[z±1])S2Vz±1S^(2)(V[z^(+-1)])S^{2}\left(V\left[z^{ \pm 1}\right]\right)S2(V[z±1]) of V[z±1]Vz±1V[z^(+-1)]V\left[z^{ \pm 1}\right]V[z±1] over kkk\mathbb{k}k, then it contains an infinity as a coefficient. To avoid such a complication, we sometimes restrict ourselves to the subgroups
G[[z]]:=G(k[[z]])⊂G((z)) and G[z]:=G(k[z])⊂G[z±1]G[[z]]:=G(k[[z]])⊂G((z)) and G[z]:=G(k[z])⊂Gz±1G[[z]]:=G(k[[z]])sub G((z))quad" and "quad G[z]:=G(k[z])sub G[z^(+-1)]G \llbracket z \rrbracket:=G(\mathbb{k} \llbracket z \rrbracket) \subset G((z)) \quad \text { and } \quad G[z]:=G(\mathbb{k}[z]) \subset G\left[z^{ \pm 1}\right]G[[z]]:=G(k[[z]])⊂G((z)) and G[z]:=G(k[z])⊂G[z±1]
We sometimes use the subgroup I⊂G[[z]]I⊂G[[z]]Isub G[[z]]\mathbf{I} \subset G \llbracket z \rrbracketI⊂G[[z]] defined by the pullback of BBBBB under the evaluation map ev0:G[[z]]→Gev0:G[[z]]→Gev_(0):G[[z]]rarr G\mathrm{ev}_{0}: G \llbracket z \rrbracket \rightarrow Gev0:G[[z]]→G at z=0z=0z=0z=0z=0. The group III\mathbf{I}I is the Iwahori subgroup obtained from (the completed version of )d)d)d) d)d by removing Gmrot Gmrot G_(m)^("rot ")\mathbb{G}_{m}^{\text {rot }}Gmrot and quotient out by the central extension.
By the quotient map k[z]→kk[z]→kk[z]rarrk\mathbb{k}[z] \rightarrow \mathbb{k}k[z]→k (and k[[z]]→kk[[z]]→kk[[z]]rarrk\mathbb{k} \llbracket z \rrbracket \rightarrow \mathbb{k}k[[z]]→k ) sending z↦0z↦0z|->0z \mapsto 0z↦0, we can regard every rational GGGGG-module VVVVV as a G[z]G[z]G[z]G[z]G[z]-module or a G[[z]]G[[z]]G[[z]]G \llbracket z \rrbracketG[[z]]-module with (trivial) Gmrot Gmrot G_(m)^("rot ")\mathbb{G}_{m}^{\text {rot }}Gmrot -action through ev0ev0ev_(0)\mathrm{ev}_{0}ev0. We also have a G[[z]]G[[z]]G[[z]]G \llbracket z \rrbracketG[[z]]-module structure (without a Gmrot Gmrot G_(m)^("rot ")\mathbb{G}_{m}^{\text {rot }}Gmrot -action) on V[[z]]:=V⊗k[[z]]V[[z]]:=V⊗k[[z]]V[[z]]:=V oxk[[z]]V \llbracket z \rrbracket:=V \otimes \mathbb{k} \llbracket z \rrbracketV[[z]]:=V⊗k[[z]] that surjects onto VVVVV.
Definition 4.1 (global Weyl modules). Let ⨀(λ)⨀(λ)⨀(lambda)\bigodot(\lambda)⨀(λ) be the category of rational G[z]G[z]G[z]G[z]G[z]-modules MMMMM that admits a decreasing filtration
M=F0M⊃F1M⊃F2M⊃⋯ such that ⋂k≥0FkM={0}M=F0M⊃F1M⊃F2M⊃⋯ such that ⋂k≥0 FkM={0}M=F_(0)M supF_(1)M supF_(2)M sup cdotsquad" such that "nnn_(k >= 0)F_(k)M={0}M=F_{0} M \supset F_{1} M \supset F_{2} M \supset \cdots \quad \text { such that } \bigcap_{k \geq 0} F_{k} M=\{0\}M=F0M⊃F1M⊃F2M⊃⋯ such that ⋂k≥0FkM={0}
and each FkM/Fk−1M(k≥1)FkM/Fk−1M(k≥1)F_(k)M//F_(k-1)M(k >= 1)F_{k} M / F_{k-1} M(k \geq 1)FkM/Fk−1M(k≥1) belongs to {qmL(μ)}m∈Z,λ≥μ∈X+qmL(μ)m∈Z,λ≥μ∈X+{q^(m)L(mu)}_(m inZ,lambda >= mu inX_(+))\left\{q^{m} L(\mu)\right\}_{m \in \mathbb{Z}, \lambda \geq \mu \in \mathbb{X}_{+}}{qmL(μ)}m∈Z,λ≥μ∈X+. For each λ∈X+λ∈X+lambda inX_(+)\lambda \in \mathbb{X}_{+}λ∈X+, we define the global Weyl module W(λ)W(λ)W(lambda)\mathbb{W}(\lambda)W(λ) of G[z]G[z]G[z]G[z]G[z] as the projective cover of L(λ)L(λ)L(lambda)L(\lambda)L(λ) in ⨀(λ)⨀(λ)⨀(lambda)\bigodot(\lambda)⨀(λ).
Note that W(λ)W(λ)W(lambda)\mathbb{W}(\lambda)W(λ) automatically acquires a Gmrot Gmrot G_(m)^("rot ")\mathbb{G}_{m}^{\text {rot }}Gmrot -action by its universality (as it exists).
Theorem 4.2. For each λ∈X+λ∈X+lambda inX_(+)\lambda \in \mathbb{X}_{+}λ∈X+with λ=∑i=1rmiϖiλ=∑i=1r miÏ–ilambda=sum_(i=1)^(r)m_(i)Ï–_(i)\lambda=\sum_{i=1}^{r} m_{i} \varpi_{i}λ=∑i=1rmiÏ–i, we have
where each xi,1,…,xi,mixi,1,…,xi,mix_(i,1),dots,x_(i,m_(i))x_{i, 1}, \ldots, x_{i, m_{i}}xi,1,…,xi,mi is of degree one with respect to the GmrotGmrotG_(m)^(rot)\mathbb{G}_{m}^{\mathrm{rot}}Gmrot-action. In addition, the action of EndG[z]W(λ)EndG[z]â¡W(λ)End_(G[z])W(lambda)\operatorname{End}_{G[z]} \mathbb{W}(\lambda)EndG[z]â¡W(λ) on W(λ)W(λ)W(lambda)\mathbb{W}(\lambda)W(λ) is free.
Theorem 4.2 was proved by Fourier-Littelmann [25] (for k=Ck=Ck=C\mathbb{k}=\mathbb{C}k=C and GGGGG of type ADE), Naoi [72] (for k=Ck=Ck=C\mathbb{k}=\mathbb{C}k=C and GGGGG of type BCFG), and it was transferred to char k>0k>0k > 0\mathbb{k}>0k>0 in [50] using results from the global bases of quantum affine algebras [4,42][4,42][4,42][4,42][4,42].
By Theorem 4.2, we factor out the positive degree parts of EndG[z]W(λ)EndG[z]â¡W(λ)End_(G[z])W(lambda)\operatorname{End}_{G[z]} \mathbb{W}(\lambda)EndG[z]â¡W(λ) to obtain
We call it a local Weyl module of G[z]G[z]G[z]G[z]G[z].
The following result clarifies that our global/local Weyl modules are the best possible analogues of Weyl modules for GGGGG (see Theorem 2.1):
Theorem 4.3 (Chari-Ion [14] for char k=0k=0k=0\mathbb{k}=0k=0, and [50] +ε+ε+epsi+\varepsilon+ε for char k>0k>0k > 0\mathbb{k}>0k>0 ). For each λ,μ∈X+λ,μ∈X+lambda,mu inX_(+)\lambda, \mu \in \mathbb{X}_{+}λ,μ∈X+, we have
The proof of Theorem 4.3 in [50,$3.3][50,$3.3][50,$3.3][50, \$ 3.3][50,$3.3] relies on the adjoint property of the Demazure functors observed in [20, PROPOSITION 5.7] and systematically utilized in [15]. The case λ=μ∗λ=μ∗lambda=mu^(**)\lambda=\mu^{*}λ=μ∗ and i>1i>1i > 1i>1i>1 in Theorem 4.3 is not recorded in [50], and might appear elsewhere.
5. SEMI-INFINITE FLAG MANIFOLDS
We keep the setting of the previous section. In view of the projectivity of W(λ)W(λ)W(lambda)\mathbb{W}(\lambda)W(λ) 's in ⨀(λ)⨀(λ)⨀(lambda)\bigodot(\lambda)⨀(λ) 's, we find unique degree-zero G[z]G[z]G[z]G[z]G[z]-module maps
with a structure of a commutative algebra compatible with the action of G[z]⋊Gmrot ×TG[z]⋊Gmrot ×TG[z]><|G_(m)^("rot ")xx TG[z] \rtimes \mathbb{G}_{m}^{\text {rot }} \times TG[z]⋊Gmrot ×T. Since the Gmrot Gmrot G_(m)^("rot ")\mathbb{G}_{m}^{\text {rot }}Gmrot -degree of RGRGR_(G)R_{G}RG is bounded from the above, the G[z]G[z]G[z]G[z]G[z]-action on RGRGR_(G)R_{G}RG automatically extends to the G[[z]]G[[z]]G[[z]]G \llbracket z \rrbracketG[[z]]-action. We set
where EEEEE is a closed subset of Spec RGRGR_(G)R_{G}RG on which the TTTTT-action is not free. Let us consider the G((z))G((z))G((z))G((z))G((z))-orbit of
viewed as a set of points, that we denote by QGQGQ_(G)\mathcal{Q}_{G}QG. By examining the coefficients of the defining relations of BBB\mathscr{B}B with its k((z))k((z))k((z))\mathbb{k}((z))k((z))-valued points, we find that the intersection
defines a closed subscheme for any choice of m1,…,mr∈Zm1,…,mr∈Zm_(1),dots,m_(r)inZm_{1}, \ldots, m_{r} \in \mathbb{Z}m1,…,mr∈Z. We denote this subscheme by QG(tβ)QGtβQ_(G)(t_(beta))\mathbf{Q}_{G}\left(t_{\beta}\right)QG(tβ), where β=∑i=1rmiαi∨β=∑i=1r miαi∨beta=sum_(i=1)^(r)m_(i)alpha_(i)^(vv)\beta=\sum_{i=1}^{r} m_{i} \alpha_{i}^{\vee}β=∑i=1rmiαi∨ is an element of the dual lattice (coroot lattice) X∨X∨X^(vv)\mathbb{X}^{\vee}X∨ of XXX\mathbb{X}X equipped with a basis {αi∨}i=1rαi∨i=1r{alpha_(i)^(vv)}_(i=1)^(r)\left\{\alpha_{i}^{\vee}\right\}_{i=1}^{r}{αi∨}i=1r such that αi∨(ϖj)=δi,jαi∨ϖj=δi,jalpha_(i)^(vv)(Ï–_(j))=delta_(i,j)\alpha_{i}^{\vee}\left(\varpi_{j}\right)=\delta_{i, j}αi∨(Ï–j)=δi,j (i.e., αi∨αi∨alpha_(i)^(vv)\alpha_{i}^{\vee}αi∨ is a simple coroot). We note that P(V(ϖi)[[z]]zmi)PVÏ–i[[z]]zmiP(V(Ï–_(i))([[)z(]])z^(m_(i)))\mathbb{P}\left(V\left(\varpi_{i}\right) \llbracket z \rrbracket z^{m_{i}}\right)P(V(Ï–i)[[z]]zmi) is a scheme, but it is not of finite type, and QG(tβ)QGtβQ_(G)(t_(beta))\mathbf{Q}_{G}\left(t_{\beta}\right)QG(tβ) is also of infinite type.
Lemma 5.1. We have QG(tβ)≅QG(tγ)QGtβ≅QGtγQ_(G)(t_(beta))~=Q_(G)(t_(gamma))\mathbf{Q}_{G}\left(t_{\beta}\right) \cong \mathbf{Q}_{G}\left(t_{\gamma}\right)QG(tβ)≅QG(tγ) for each pair β,γ∈X∨β,γ∈X∨beta,gamma inX^(vv)\beta, \gamma \in \mathbb{X}^{\vee}β,γ∈X∨ as schemes equipped with G[[z]]G[[z]]G[[z]]G \llbracket z \rrbracketG[[z]]-actions. Hence, the union
is a pure ind-scheme of ind-infinite type equipped with the action of G[[z]]⋊GmrotG[[z]]⋊GmrotG[[z]]><|G_(m)^(rot)G \llbracket z \rrbracket \rtimes \mathbb{G}_{m}^{\mathrm{rot}}G[[z]]⋊Gmrot. Moreover, the set of G[[z]]G[[z]]G[[z]]G \llbracket z \rrbracketG[[z]]-orbits in QGrat QGrat Q_(G)^("rat ")\mathbf{Q}_{G}^{\text {rat }}QGrat is in bijection with X∨X∨X^(vv)\mathbb{X}^{\vee}X∨.
In effect, we have an open dense G[[z]]G[[z]]G[[z]]G \llbracket z \rrbracketG[[z]]-orbit OG(tβ)⊂QG(tβ)OGtβ⊂QGtβO_(G)(t_(beta))subQ_(G)(t_(beta))\mathbf{O}_{G}\left(t_{\beta}\right) \subset \mathbf{Q}_{G}\left(t_{\beta}\right)OG(tβ)⊂QG(tβ) that is isomorphic to G[[z]]/(T⋅N[[z]])G[[z]]/(Tâ‹…N[[z]])G[[z]]//(T*N[[z]])G \llbracket z \rrbracket /(T \cdot N \llbracket z \rrbracket)G[[z]]/(Tâ‹…N[[z]]). By the Bruhat decomposition, we divide OG(tβ)OGtβO_(G)(t_(beta))\mathbf{O}_{G}\left(t_{\beta}\right)OG(tβ) into the disjoint union of III\mathbf{I}I-orbits as ⨆w∈WO(wtβ)⨆w∈W Owtβ⨆_(w in W)O(wt_(beta))\bigsqcup_{w \in W} \mathbf{O}\left(w t_{\beta}\right)⨆w∈WO(wtβ) such that O(tβ)⊂OG(tβ)Otβ⊂OGtβO(t_(beta))subO_(G)(t_(beta))\mathbf{O}\left(t_{\beta}\right) \subset \mathbf{O}_{G}\left(t_{\beta}\right)O(tβ)⊂OG(tβ) is open dense. Identifying β∈X∨β∈X∨beta inXvv\beta \in \mathbb{X} \veeβ∈X∨ with tβtβt_(beta)t_{\beta}tβ, we set Waf:=W⋉X∨Waf:=W⋉X∨W_(af):=W|><X^(vv)W_{\mathrm{af}}:=W \ltimes \mathbb{X}^{\vee}Waf:=W⋉X∨. We define
QG(w):=O(w)¯⊂QGrat,w∈WafQG(w):=O(w)¯⊂QGrat,w∈WafQ_(G)(w):= bar(O(w))subQ_(G)^(rat),quad w inW_(af)\mathbf{Q}_{G}(w):=\overline{\mathbf{O}(w)} \subset \mathbf{Q}_{G}^{\mathrm{rat}}, \quad w \in W_{\mathrm{af}}QG(w):=O(w)¯⊂QGrat,w∈Waf
The inclusion relation on {QG(w)}w∈WafQG(w)w∈Waf{Q_(G)(w)}_(w inW_(af))\left\{\mathbf{Q}_{G}(w)\right\}_{w \in W_{\mathrm{af}}}{QG(w)}w∈Waf is described by the generic Bruhat order [62]. We refer to the partial order on Waf Waf W_("af ")W_{\text {af }}Waf induced from this closure ordering by ≤∞2≤∞2<= (oo)/(2)\leq \frac{\infty}{2}≤∞2 as in [50,52][50,52][50,52][50,52][50,52] (there we sometimes called ≤∞2≤∞2<= (oo)/(2)\leq \frac{\infty}{2}≤∞2 as the semi-infinite Bruhat order).
Theorem 5.2. The scheme QG(w)QG(w)Q_(G)(w)\mathbf{Q}_{G}(w)QG(w) is normal for each w∈Wafw∈Wafw inW_(af)w \in W_{\mathrm{af}}w∈Waf. In addition, the ind-scheme QGrat QGrat Q_(G)^("rat ")\mathbf{Q}_{G}^{\text {rat }}QGrat is a strict ind-scheme in the sense that each inclusion is a closed immersion. The indscheme QGrat QGrat Q_(G)^("rat ")\mathbf{Q}_{G}^{\text {rat }}QGrat coarsely ind-represents the coset G((z))/(T⋅N((z)))G((z))/(Tâ‹…N((z)))G((z))//(T*N((z)))G((z)) /(T \cdot N((z)))G((z))/(Tâ‹…N((z))).
The first two statements are proved in [52] when char k=0k=0k=0k=0k=0. The proof valid for char k≠2k≠2k!=2\mathbb{k} \neq 2k≠2, as well as the last assertion, are contained in [50]. This last assertion says that the (ind-)scheme QGrat QGrat Q_(G)^("rat ")\mathbf{Q}_{G}^{\text {rat }}QGrat is the universal one that maps to every (ind-)scheme whose points yield QGQGQ_(G)Q_{G}QG. It follows that if we take a family {Y(λ)}λ∈X+{Y(λ)}λ∈X+{Y(lambda)}_(lambda inX_(+))\{\mathbb{Y}(\lambda)\}_{\lambda \in \mathbb{X}_{+}}{Y(λ)}λ∈X+instead of {W(λ)}λ∈X+{W(λ)}λ∈X+{W(lambda)}_(lambda inX_(+))\{\mathbb{W}(\lambda)\}_{\lambda \in \mathbb{X}_{+}}{W(λ)}λ∈X+to define QG(tβ)QGtβQ_(G)(t_(beta))\mathbf{Q}_{G}\left(t_{\beta}\right)QG(tβ), then the corresponding coordinate ring RG′RG′R_(G)^(')R_{G}^{\prime}RG′ admits a map to RGRGR_(G)R_{G}RG. Let us point out that this can be thought of as a family version of the properties of global Weyl modules discussed in Section 4, and we indeed have several reasonable choices of {Y(λ)}λ∈X+{Y(λ)}λ∈X+{Y(lambda)}_(lambda inX_(+))\{\mathbb{Y}(\lambda)\}_{\lambda \in \mathbb{X}_{+}}{Y(λ)}λ∈X+other than {W(λ)}λ∈X+{W(λ)}λ∈X+{W(lambda)}_(lambda inX_(+))\{\mathbb{W}(\lambda)\}_{\lambda \in \mathbb{X}_{+}}{W(λ)}λ∈X+including the coordinate ring of the arc scheme of G/NG/NG//NG / NG/N. For simplicity, we may refer to QG(t0)QGt0Q_(G)(t_(0))\mathbf{Q}_{G}\left(t_{0}\right)QG(t0) as QGQGQ_(G)\mathbf{Q}_{G}QG below.
The inclusion
(5.4)QG⊂∏i=1rP(V(ϖi)[[z]])(5.4)QG⊂âˆi=1r PVÏ–i[[z]]{:(5.4)Q_(G)subprod_(i=1)^(r)P(V(Ï–_(i))([[)z(]])):}\begin{equation*}
\mathbf{Q}_{G} \subset \prod_{i=1}^{r} \mathbb{P}\left(V\left(\varpi_{i}\right) \llbracket z \rrbracket\right) \tag{5.4}
\end{equation*}(5.4)QG⊂âˆi=1rP(V(Ï–i)[[z]])
induces a line bundle OQG(ϖi)OQGÏ–iO_(Q_(G))(Ï–_(i))\mathcal{O}_{\mathbf{Q}_{G}}\left(\varpi_{i}\right)OQG(Ï–i) on QGQGQ_(G)\mathbf{Q}_{G}QG, that is, the pull-back of O(1)O(1)O(1)\mathcal{O}(1)O(1) from P(V(ϖi)[[z]])PVÏ–i[[z]]P(V(Ï–_(i))([[)z(]]))\mathbb{P}\left(V\left(\varpi_{i}\right) \llbracket z \rrbracket\right)P(V(Ï–i)[[z]]). By taking the tensor products, we have OQG(λ):=⨂i=1rOQG(ϖi)⊗niOQG(λ):=⨂i=1r OQGÏ–i⊗niO_(Q_(G))(lambda):=⨂_(i=1)^(r)O_(Q_(G))(Ï–_(i))^(oxn_(i))\mathcal{O}_{\mathbf{Q}_{G}}(\lambda):=\bigotimes_{i=1}^{r} \mathcal{O}_{\mathbf{Q}_{G}}\left(\varpi_{i}\right)^{\otimes n_{i}}OQG(λ):=⨂i=1rOQG(Ï–i)⊗ni for λ=∑i=1rniϖiλ=∑i=1r niÏ–ilambda=sum_(i=1)^(r)n_(i)Ï–_(i)\lambda=\sum_{i=1}^{r} n_{i} \varpi_{i}λ=∑i=1rniÏ–i(ni∈Z)ni∈Z(n_(i)inZ)\left(n_{i} \in \mathbb{Z}\right)(ni∈Z). By Lemma 5.1, we have OQGrat (λ)(λ∈X)OQGrat (λ)(λ∈X)O_(Q_(G)^("rat "))(lambda)(lambda inX)\mathcal{O}_{\mathbf{Q}_{G}^{\text {rat }}}(\lambda)(\lambda \in \mathbb{X})OQGrat (λ)(λ∈X) on QGrat QGrat Q_(G)^("rat ")\mathbf{Q}_{G}^{\text {rat }}QGrat that yields OQG(λ)OQG(λ)O_(Q_(G))(lambda)\mathcal{O}_{\mathbf{Q}_{G}}(\lambda)OQG(λ) by restriction.
Theorem 5.3 ([52] for char k=0k=0k=0\mathbb{k}=0k=0, and [50] for char k≠2k≠2k!=2\mathbb{k} \neq 2k≠2 ). For each λ∈Xλ∈Xlambda inX\lambda \in \mathbb{X}λ∈X, we have
The proof of Theorem 5.3 depends on the freeness of RGRGR_(G)R_{G}RG over an infinitely-manyvariable polynomial ring, that yields a regular sequence of infinite length. Such a situation never occur for finite type schemes, or infinite type schemes like BCthick BCthick B_(C)^("thick ")\mathscr{B}_{C}^{\text {thick }}BCthick . In case G=SL(2)G=SLâ¡(2)G=SL(2)G=\operatorname{SL}(2)G=SLâ¡(2), Theorem 5.3 reduces to an exercise in algebraic geometry by QG≅P(k2[[z]])QG≅Pk2[[z]]Q_(G)~=P(k^(2)([[)z(]]))\mathbf{Q}_{G} \cong \mathbb{P}\left(\mathbb{k}^{2} \llbracket z \rrbracket\right)QG≅P(k2[[z]]).
Theorem 5.3 has an ind-model counterpart proved earlier [10]. The Frobenius splitting of QGQGQ_(G)\mathbf{Q}_{G}QG (explained later) and Theorem 5.3 imply this ind-model counterpart. However, the author is uncertain whether [10] implies Theorem 5.3 (even in case char k=0k=0k=0k=0k=0 ) since the natural ring coming from the ind-model is a completion of RGRGR_(G)R_{G}RG, and the completion operation of a ring loses information in general. We have an analogue of Theorem 5.3 for all III\mathbf{I}I-orbit closures, proved for the ind-model in [46,50] and for the formal model in [50,52].
6. FROBENIUS SPLITTINGS
We continue to work in the setting of the previous section. We fix a prime p>0p>0p > 0p>0p>0. For a scheme XXX\mathfrak{X}X over FpFpF_(p)\mathbb{F}_{p}Fp, we have a Frobenius morphism Fr:X→XFr:X→XFr:XrarrX\mathrm{Fr}: \mathfrak{X} \rightarrow \mathfrak{X}Fr:X→X induced from the ppppp th power map. We have a natural map Fr∗OX→OXFr∗â¡OX→OXFr^(**)O_(X)rarrO_(X)\operatorname{Fr}^{*} \mathcal{O}_{\mathfrak{X}} \rightarrow \mathcal{O}_{\mathfrak{X}}Fr∗â¡OX→OX that induces a map OX→Fr∗OXOX→Fr∗OXO_(X)rarrFr_(**)O_(X)\mathcal{O}_{\mathfrak{X}} \rightarrow \mathrm{Fr}_{*} \mathcal{O}_{\mathfrak{X}}OX→Fr∗OX by adjunction. The Frobenius splitting ϕ:Fr∗OX→OXÏ•:Fr∗â¡OX→OXphi:Fr_(**)O_(X)rarrO_(X)\phi: \operatorname{Fr}_{*} \mathcal{O}_{\mathfrak{X}} \rightarrow \mathcal{O}_{\mathfrak{X}}Ï•:Fr∗â¡OX→OX is an OX-module map such that the composition OX-module map such that the composition O_(X"-module map such that the composition ")\mathcal{O}_{\mathfrak{X} \text {-module map such that the composition }}OX-module map such that the compositionÂ
is the identity. If XXX\mathfrak{X}X is projective (and is of finite type) and OXOXO_(X)\mathcal{O}_{\mathfrak{X}}OX admits a Frobenius splitting, then XXX\mathfrak{X}X is reduced and an ample line bundle has the higher cohomology vanishing [68].
For generality on Frobenius splittings, as well as their applications to BBB\mathscr{B}B and BCthin BCthin B_(C)^("thin ")\mathscr{B}_{C}^{\text {thin }}BCthin , we refer to Brion-Kumar [12] (note that [12] has a finite type assumption, that we drop in case the proof does not require it. In the paragraph above, reducedness does not require the finite type assumption, while the higher cohomology vanishing requires the finite type assumption through the Serre vanishing). Frobenius splitting of BBB\mathscr{B}B in char k=pk=pk=p\mathbb{k}=pk=p is useful in proving that Schubert and Richardson varieties are reduced, normal, and have rational singularities. There are two major ways to construct a Frobenius splitting of BBB\mathscr{B}B : one is to investigate the global section of the (1−p)(1−p)(1-p)(1-p)(1−p) th power of the canonical bundle, and the other is to use a Bott-Samelson-Demazure-Hansen (=BSDH) resolution of BBB\mathscr{B}B.
Since BCthin BCthin B_(C)^("thin ")\mathscr{B}_{C}^{\text {thin }}BCthin is no longer smooth, we cannot use the canonical bundle to construct a Frobenius splitting. Nevertheless, a (partial) BSDH resolution does the job. The situation of BCthick BCthick B_(C)^("thick ")\mathscr{B}_{C}^{\text {thick }}BCthick is a bit worse. The canonical bundle of BCthick BCthick B_(C)^("thick ")\mathscr{B}_{C}^{\text {thick }}BCthick makes some sense, but the author does not know whether it has enough power to produce a Frobenius splitting. The scheme BCthick BCthick B_(C)^("thick ")\mathscr{B}_{C}^{\text {thick }}BCthickÂ
admits a BSDH resolution, but it is a successive P1P1P^(1)\mathbb{P}^{1}P1-fibration over an infinite-type scheme. Thus, we cannot equip BCthick BCthick B_(C)^("thick ")\mathscr{B}_{C}^{\text {thick }}BCthick with a Frobenius splitting by either of the above means at present. Despite this, we can transfer a Frobenius splitting of BCthin BCthin B_(C)^("thin ")\mathscr{B}_{C}^{\text {thin }}BCthin to BCthick BCthick B_(C)^("thick ")\mathscr{B}_{C}^{\text {thick }}BCthick by using the compatible splitting property of a point [49], following an idea of Mathieu.
Frobenius splitting of QGrat QGrat Q_(G)^("rat ")\mathbf{Q}_{G}^{\text {rat }}QGrat (or rather each of its ind-piece QG(w)QG(w)Q_(G)(w)\mathbf{Q}_{G}(w)QG(w) ) is used below, and hence we need a recipe to produce one. However, the situation of the BSDH resolution is similar to that of BCthick BCthick B_(C)^("thick ")\mathscr{B}_{C}^{\text {thick }}BCthick , and the canonical bundle on QGrat QGrat Q_(G)^("rat ")\mathbf{Q}_{G}^{\text {rat }}QGrat simply does not make sense naively (e.g., its TTTTT-weight at a point must be infinity). Therefore, we need a new proof strategy. Our strategy in [50] is to regard RGRGR_(G)R_{G}RG as a subalgebra of the corresponding coordinate ring of BCthick BCthick B_(C)^("thick ")\mathscr{B}_{C}^{\text {thick }}BCthick , and prove that a Frobenius splitting of BCthick BCthick B_(C)^("thick ")\mathscr{B}_{C}^{\text {thick }}BCthick preserves RGRGR_(G)R_{G}RG. For this, we first see that each W(mλ)(m∈Z>0,λ∈X+)W(mλ)m∈Z>0,λ∈X+W(m lambda)(m inZ_( > 0),lambda inX_(+))\mathbb{W}(m \lambda)\left(m \in \mathbb{Z}_{>0}, \lambda \in \mathbb{X}_{+}\right)W(mλ)(m∈Z>0,λ∈X+)is a quotient of L(mΛ)L(mΛ)L(m Lambda)L(m \Lambda)L(mΛ) for some Λ∈P+Λ∈P+Lambda inP_(+)\Lambda \in P_{+}Λ∈P+by twisting the G[z−1]Gz−1G[z^(-1)]G\left[z^{-1}\right]G[z−1]-action into a G[z]G[z]G[z]G[z]G[z]-action as z−1↦zz−1↦zz^(-1)|->zz^{-1} \mapsto zz−1↦z. Let πm:L(mΛ)→W(mλ)Ï€m:L(mΛ)→W(mλ)pi_(m):L(m Lambda)rarrW(m lambda)\pi_{m}: L(m \Lambda) \rightarrow \mathbb{W}(m \lambda)Ï€m:L(mΛ)→W(mλ) be the quotient map. This embeds (a suitable ZZZ\mathbb{Z}Z-graded subalgebra of) RGRGR_(G)R_{G}RG into (3.1) as an algebra with G[[z]]⋉Gmrot G[[z]]⋉Gmrot G[[z]]|><G_(m)^("rot ")G \llbracket z \rrbracket \ltimes \mathbb{G}_{m}^{\text {rot }}G[[z]]⋉Gmrot -action. We need to show that the map ϕ∨ϕ∨phi^(vv)\phi^{\vee}ϕ∨ obtained by dualizing the Frobenius splitting of BCthick BCthick B_(C)^("thick ")\mathscr{B}_{C}^{\text {thick }}BCthick induces a map ϕW∨Ï•W∨phi_(W)^(vv)\phi_{\mathbb{W}}^{\vee}Ï•W∨ in the following diagram:
This is equivalent to seeing that ϕ∨(kerπm)⊂kerπpmϕ∨kerâ¡Ï€m⊂kerâ¡Ï€pmphi^(vv)(ker pi_(m))sub ker pi_(pm)\phi^{\vee}\left(\operatorname{ker} \pi_{m}\right) \subset \operatorname{ker} \pi_{p m}ϕ∨(kerâ¡Ï€m)⊂kerâ¡Ï€pm. We use the projectivity of W(mλ)W(mλ)W(m lambda)\mathbb{W}(m \lambda)W(mλ) in ⨀(mλ)⨀(mλ)⨀(m lambda)\bigodot(m \lambda)⨀(mλ) to assume that the G[z]G[z]G[z]G[z]G[z]-module generators of ker πmÏ€mpi_(m)\pi_{m}Ï€m have TTTTT-weights that do not appear in W(mλ)W(mλ)W(m lambda)\mathbb{W}(m \lambda)W(mλ). In view of the fact that ker πpmÏ€pmpi_(pm)\pi_{p m}Ï€pm contains all the TTTTT-weight spaces in L(pmΛ)L(pmΛ)L(pm Lambda)L(p m \Lambda)L(pmΛ) whose TTTTT-weights do not appear in W(pmλ)W(pmλ)W(pm lambda)\mathbb{W}(p m \lambda)W(pmλ), we have necessarily ϕ∨(kerπm)⊂ϕ∨kerâ¡Ï€m⊂phi^(vv)(ker pi_(m))sub\phi^{\vee}\left(\operatorname{ker} \pi_{m}\right) \subsetϕ∨(kerâ¡Ï€m)⊂kerπpmkerâ¡Ï€pmker pi_(pm)\operatorname{ker} \pi_{p m}kerâ¡Ï€pm by the TTTTT-weight comparison of the generators.
In fact, every L(Λ)L(Λ)L(Lambda)L(\Lambda)L(Λ) admits a filtration by global Weyl modules when char k=0k=0k=0\mathbb{k}=0k=0 if we twist the action of G[z]G[z]G[z]G[z]G[z] on global Weyl modules into G[z−1][51]Gz−1[51]G[z^(-1)][51]G\left[z^{-1}\right][51]G[z−1][51]. Therefore, we indeed obtain a Frobenius splitting of QGQGQ_(G)\mathbf{Q}_{G}QG via a novel proof based on the "universality" of the global Weyl module W(λ)W(λ)W(lambda)\mathbb{W}(\lambda)W(λ) explained in Section 4. In conclusion, we have:
Theorem 6.1 ([50, THeORem B]). The ind-scheme QGrat QGrat Q_(G)^("rat ")\mathbf{Q}_{G}^{\text {rat }}QGrat admits a Frobenius splitting that is compatible with all III\mathbf{I}I-orbits when char k>2k>2k > 2\mathbb{k}>2k>2.
7. CONNECTION TO THE SPACE OF RATIONAL MAPS
Keep the setting as in Section 5. Let us consider the vector space embedding k((z))⊂k[[z,z−1]]k((z))⊂k[[z,z−1]]k((z))subk[[z,z^(-1)]]\mathbb{k}((z)) \subset \mathbb{k} \llbracket z, z^{-1} \rrbracketk((z))⊂k[[z,z−1]] into the formal power series with unbounded powers. The space k[[z,z−1]]k[[z,z−1]]k[[z,z^(-1)]]\mathbb{k} \llbracket z, z^{-1} \rrbracketk[[z,z−1]] no longer forms a ring. Nevertheless, we have an automorphism of k[[z,z−1]]k[[z,z−1]]k[[z,z^(-1)]]\mathbb{k} \llbracket z, z^{-1} \rrbracketk[[z,z−1]] by swapping zzzzz with z−1z−1z^(-1)z^{-1}z−1. Together with the Chevalley involution of GGGGG (an automorphism of GGGGG that sends each element of TTTTT to its inverse), it induces an involution θθtheta\thetaθ on the ambient space
The scheme QG(w,v)QG(w,v)Q_(G)(w,v)\mathcal{Q}_{G}(w, v)QG(w,v) is always of finite type, and the case w,v∈Ww,v∈Ww,v in Ww, v \in Ww,v∈W yields a Richardson variety of BBB\mathscr{B}B. The normality part of the proof of Theorem 7.1 goes as follows: Our Frobenius splitting of QGrat QGrat Q_(G)^("rat ")\mathbf{Q}_{G}^{\text {rat }}QGrat induces a Frobenius splitting of QG(w,v)QG(w,v)Q_(G)(w,v)\mathcal{Q}_{G}(w, v)QG(w,v). In particular, it is reduced and weakly normal in char k>2k>2k > 2\mathbb{k}>2k>2. (Here a weakly normal ring is essentially a normal ring up to topology.) Then, we lift the weak normality to characteristic zero and prove the normality of the intersection by a geometric consideration. Once we deduce the normality in characteristic zero, we can reduce it to char k≫0k≫0k≫0\mathbb{k} \gg 0k≫0 by a general result.
Let us exhibit some relevant geometric considerations here. To this end, we assume k=Ck=Ck=C\mathbb{k}=\mathbb{C}k=C in the rest of this section. Recall that H2(B,Z)≅X∨H2(B,Z)≅X∨H_(2)(B,Z)~=X^(vv)H_{2}(\mathscr{B}, \mathbb{Z}) \cong \mathbb{X}^{\vee}H2(B,Z)≅X∨. Let EB2,βEB2,βEB_(2,beta)\mathcal{E} \mathscr{B}_{2, \beta}EB2,β (resp. B2,β)B2,β{:B_(2,beta))\left.\mathscr{B}_{2, \beta}\right)B2,β) be the space of genus-zero stable maps with two marked points to (P1×B)(P1×B((P^(1)xxB)(\left(\mathbb{P}^{1} \times \mathscr{B}\right)((P1×B)( resp. B)B)B)\mathscr{B})B) whose image has class (1,β)∈H2(P1×B,Z)(1,β)∈H2P1×B,Z(1,beta)inH_(2)(P^(1)xxB,Z)(1, \beta) \in H_{2}\left(\mathbb{P}^{1} \times \mathscr{B}, \mathbb{Z}\right)(1,β)∈H2(P1×B,Z) (resp. β∈H2(B,Z)β∈H2(B,Z)beta inH_(2)(B,Z)\beta \in H_{2}(\mathscr{B}, \mathbb{Z})β∈H2(B,Z) ), regarded as an algebraic variety with rational singularities [28]. We have a subvariety EB2,βbEB2,βbEB_(2,beta)^(b)\mathscr{E} \mathscr{B}_{2, \beta}^{b}EB2,βb such that the first marked point lands in 0∈P10∈P10inP^(1)0 \in \mathbb{P}^{1}0∈P1 and the second marked point lands in ∞∈P1∞∈P1oo inP^(1)\infty \in \mathbb{P}^{1}∞∈P1 through the composition
Consider the Schubert variety (a BBBBB-orbit closure) B(w)⊂BB(w)⊂BB(w)subB\mathscr{B}(w) \subset \mathscr{B}B(w)⊂B corresponding to w∈Ww∈Ww in Ww \in Ww∈W and the opposite Schubert variety (a θ(B)θ(B)theta(B)\theta(B)θ(B)-orbit closure) Bop(v)⊂BBop(v)⊂BB^(op)(v)subB\mathscr{B}^{\mathrm{op}}(v) \subset \mathscr{B}Bop(v)⊂B corresponding to v∈Wv∈Wv in Wv \in Wv∈W.
Let QG(β)(β∈X∨)QG(β)β∈X∨Q_(G)(beta)(beta inX^(vv))\mathcal{Q}_{G}(\beta)\left(\beta \in \mathbb{X}^{\vee}\right)QG(β)(β∈X∨) denote the space of quasimaps from P1P1P^(1)\mathbb{P}^{1}P1 to BBB\mathscr{B}B of degree ββbeta\betaβ [22], that is, a natural compactification of Q∘G(β)Q∘G(β)Q^(@)_(G)(beta)\stackrel{\circ}{Q}_{G}(\beta)Q∘G(β) such that
where γ≤βγ≤βgamma <= beta\gamma \leq \betaγ≤β is defined as β−γ∈∑i=1rZ≥0αi∨β−γ∈∑i=1r Z≥0αi∨beta-gamma insum_(i=1)^(r)Z_( >= 0)alpha_(i)^(vv)\beta-\gamma \in \sum_{i=1}^{r} \mathbb{Z}_{\geq 0} \alpha_{i}^{\vee}β−γ∈∑i=1rZ≥0αi∨, and
(P1)γ=∏i=1r((P1)mi/Smi) where γ=∑i=1rmiαi∨P1γ=âˆi=1r P1mi/Smi where γ=∑i=1r miαi∨(P^(1))^(gamma)=prod_(i=1)^(r)((P^(1))^(m_(i))//S_(m_(i)))quad" where "gamma=sum_(i=1)^(r)m_(i)alpha_(i)^(vv)\left(\mathbb{P}^{1}\right)^{\gamma}=\prod_{i=1}^{r}\left(\left(\mathbb{P}^{1}\right)^{m_{i}} / \mathbb{S}_{m_{i}}\right) \quad \text { where } \gamma=\sum_{i=1}^{r} m_{i} \alpha_{i}^{\vee}(P1)γ=âˆi=1r((P1)mi/Smi) where γ=∑i=1rmiαi∨
Here (P1)γP1γ(P^(1))^(gamma)\left(\mathbb{P}^{1}\right)^{\gamma}(P1)γ records the place where the degree of the genuine map drops in which degree components (without ordering). By adding extra P1P1P^(1)\mathbb{P}^{1}P1 components and (compatible) maps to BBB\mathscr{B}B to P1P1P^(1)\mathbb{P}^{1}P1 in (f:P1→B)∈Q∘G(β−γ)f:P1→B∈Q∘G(β−γ)(f:P^(1)rarrB)inQ^(@)_(G)(beta-gamma)\left(f: \mathbb{P}^{1} \rightarrow \mathscr{B}\right) \in \stackrel{\circ}{Q}_{G}(\beta-\gamma)(f:P1→B)∈Q∘G(β−γ) at the places (and total degrees) recorded by (P1)γP1γ(P^(1))^(gamma)\left(\mathbb{P}^{1}\right)^{\gamma}(P1)γ (for each 0≤γ≤β0≤γ≤β0 <= gamma <= beta0 \leq \gamma \leq \beta0≤γ≤β ), we obtain a map of topological spaces
that is an identity on Q∘G(β)Q∘G(β)Q^(@)_(G)(beta)\stackrel{\circ}{Q}_{G}(\beta)Q∘G(β). Givental's main lemma asserts that this is a birational morphism of integral algebraic varieties.
Proposition 7.2 ( [50,$5.2])[50,$5.2])[50,$5.2])[50, \$ 5.2])[50,$5.2]). For each β∈X∨β∈X∨beta inX^(vv)\beta \in \mathbb{X}^{\vee}β∈X∨, we have
as schemes. In addition, πÏ€pi\piÏ€ restricts to a birational morphism
πβ,w,v:EBβ(w,v)→QG(w,vtβ),w,v∈Wπβ,w,v:EBβ(w,v)→QGw,vtβ,w,v∈Wpi_(beta,w,v):EB_(beta)(w,v)rarrQ_(G)(w,vt_(beta)),quad w,v in W\pi_{\beta, w, v}: \mathscr{E} \mathscr{B}_{\beta}(w, v) \rightarrow \mathbb{Q}_{G}\left(w, v t_{\beta}\right), \quad w, v \in Wπβ,w,v:EBβ(w,v)→QG(w,vtβ),w,v∈W
In particular, we have GBβ(w,v)≠∅GBβ(w,v)≠∅GB_(beta)(w,v)!=O/\mathscr{G} \mathscr{B}_{\beta}(w, v) \neq \emptysetGBβ(w,v)≠∅ if and only if w≤∞2vtβw≤∞2vtβw <= (oo)/(2)vt_(beta)w \leq \frac{\infty}{2} v t_{\beta}w≤∞2vtβ, and its dimension is given by the distance between wwwww and vtβvtβvt_(beta)v t_{\beta}vtβ with respect to ≤∞2≤∞2<= (oo)/(2)\leq \frac{\infty}{2}≤∞2.
In other words, the Richardson varieties of QGrat QGrat Q_(G)^("rat ")\mathbf{Q}_{G}^{\text {rat }}QGrat are precisely the spaces of quasimaps, possibly with additional conditions imposed by the space of stable maps. According to Buch-Chaput-Mihalcea-Perrin [13], the variety EBβ(w,v)EBβ(w,v)EB_(beta)(w,v)\mathscr{E} \mathscr{B}_{\beta}(w, v)EBβ(w,v) is irreducible and has rational singularities if it is nonempty. Hence, we find that QG(w,vtβ)QGw,vtβQ_(G)(w,vt_(beta))\mathcal{Q}_{G}\left(w, v t_{\beta}\right)QG(w,vtβ) is irreducible in general. Proposition 7.2 and properties of the maps πβ,w,vπβ,w,vpi_(beta,w,v)\pi_{\beta, w, v}πβ,w,v are used in our proof of Theorem 7.1.
Proposition 7.2 implies that QG(w,vtβ)QGw,vtβQ_(G)(w,vt_(beta))\mathcal{Q}_{G}\left(w, v t_{\beta}\right)QG(w,vtβ) is the closure (in QG(β)QG(β)Q_(G)(beta)\mathcal{Q}_{G}(\beta)QG(β) ) of the space of maps from P1P1P^(1)\mathbb{P}^{1}P1 to BBB\mathscr{B}B such that 0,∞∈P10,∞∈P10,oo inP^(1)0, \infty \in \mathbb{P}^{1}0,∞∈P1 land in B(w)B(w)B(w)\mathscr{B}(w)B(w) and B∘p(v)B∘p(v)B^(@p)(v)\mathscr{B}^{\circ p}(v)B∘p(v), respectively. By examining the natural map EBβ(w,v)→Bβ(w,v)EBβ(w,v)→Bβ(w,v)EB_(beta)(w,v)rarrB_(beta)(w,v)\mathscr{E} \mathscr{B}_{\beta}(w, v) \rightarrow \mathscr{B}_{\beta}(w, v)EBβ(w,v)→Bβ(w,v) (obtained by forgetting the map to P1P1P^(1)\mathbb{P}^{1}P1 ), we obtain:
Corollary 7.3. For all w,v∈Ww,v∈Ww,v in Ww, v \in Ww,v∈W and 0≠β∈X∨0≠β∈X∨0!=beta inX^(vv)0 \neq \beta \in \mathbb{X}^{\vee}0≠β∈X∨, we have
dimBβ(w,v)=dimEBβ(w,v)−1 if EBβ(w,v)≠∅dimâ¡Bβ(w,v)=dimâ¡EBβ(w,v)−1 if EBβ(w,v)≠∅dim B_(beta)(w,v)=dim EB_(beta)(w,v)-1quad" if "EB_(beta)(w,v)!=O/\operatorname{dim} \mathscr{B}_{\beta}(w, v)=\operatorname{dim} \mathscr{E} \mathscr{B}_{\beta}(w, v)-1 \quad \text { if } \mathscr{E} \mathscr{B}_{\beta}(w, v) \neq \emptysetdimâ¡Bβ(w,v)=dimâ¡EBβ(w,v)−1 if EBβ(w,v)≠∅
and Bβ(w,v)≠∅Bβ(w,v)≠∅B_(beta)(w,v)!=O/\mathscr{B}_{\beta}(w, v) \neq \emptysetBβ(w,v)≠∅ if and only if EBβ(w,v)≠∅EBβ(w,v)≠∅EB_(beta)(w,v)!=O/\mathcal{E} \mathscr{B}_{\beta}(w, v) \neq \emptysetEBβ(w,v)≠∅. Moreover, we have
Bβ(w,v)≠∅ and dimBβ(w,v)=0Bβ(w,v)≠∅ and dimâ¡Bβ(w,v)=0B_(beta)(w,v)!=O/quad" and "quad dim B_(beta)(w,v)=0\mathscr{B}_{\beta}(w, v) \neq \emptyset \quad \text { and } \quad \operatorname{dim} \mathscr{B}_{\beta}(w, v)=0Bβ(w,v)≠∅ and dimâ¡Bβ(w,v)=0
if and only if w≤∞2vtβw≤∞2vtβw <= (oo)/(2)vt_(beta)w \leq \frac{\infty}{2} v t_{\beta}w≤∞2vtβ are adjacent with respect to ≤∞2≤∞2<= (oo)/(2)\leq \frac{\infty}{2}≤∞2. In such a case, Bβ(w,v)Bβ(w,v)B_(beta)(w,v)\mathscr{B}_{\beta}(w, v)Bβ(w,v) is a point.
Thanks to the dimension axiom in quantum correlators [54, (2.5)], Corollary 7.3 describes which (primary) two-point cohomological Gromov-Witten invariant of BBB\mathscr{B}B with respect to the Schubert bases is nonzero (we can also tell its exact value). By the divisor axiom [54, $2.2.4] and the classical Chevalley formula [16], we find the Chevalley formula in quantum cohomology of BBB\mathscr{B}B from this [29]. This clarifies the role of QG(w,vtβ)QGw,vtβQ_(G)(w,vt_(beta))\mathcal{Q}_{G}\left(w, v t_{\beta}\right)QG(w,vtβ) in the study of quantum cohomology of BBB\mathscr{B}B from our perspective.
Theorem 7.4 ([47]). Let β∈X∨β∈X∨beta inX^(vv)\beta \in \mathbb{X}^{\vee}β∈X∨ and w,v∈Ww,v∈Ww,v in Ww, v \in Ww,v∈W. The variety QG(w,vtβ)QGw,vtβQ_(G)(w,vt_(beta))Q_{G}\left(w, v t_{\beta}\right)QG(w,vtβ) has rational singularities.
Theorem 7.4 is proved by Braverman-Finkelberg [9,10][9,10][9,10][9,10][9,10] for the case w=e,v=w0w=e,v=w0w=e,v=w_(0)w=e, v=w_{0}w=e,v=w0 by an analysis of Zastava spaces, which does not extend to general w,vw,vw,vw, vw,v. Theorem 7.4 is the most subtle technical point in [47] and its induction steps become possible by Theorem 7.1.
8. KKKKK-THEORETIC PETERSON ISOMORPHISM
We follow the setting of the previous section with k=Ck=Ck=C\mathbb{k}=\mathbb{C}k=C. We understand that the KKKKK-groups appearing here contain a suitable class of line bundles supported on subvarieties equipped with some group actions, and its scalar is extended from ZZZ\mathbb{Z}Z to CCC\mathbb{C}C. Let GrG:=G((z))/G[[z]]GrG:=G((z))/G[[z]]Gr_(G):=G((z))//G[[z]]\operatorname{Gr}_{G}:=G((z)) / G \llbracket z \rrbracketGrG:=G((z))/G[[z]] be the affine Grassmannian of GGGGG. The set of III\mathbf{I}I-orbits in GrGGrGGr_(G)\mathrm{Gr}_{G}GrG is in bijection with X∨X∨X^(vv)\mathbb{X}^{\vee}X∨, while the set of G[[z]]G[[z]]G[[z]]G \llbracket z \rrbracketG[[z]]-orbits of GrGGrGGr_(G)\mathrm{Gr}_{G}GrG is in bijection with X<∨⊂X∨X<∨⊂X∨X_( < )^(vv)subX^(vv)\mathbb{X}_{<}^{\vee} \subset \mathbb{X}^{\vee}X<∨⊂X∨ formed by the set of antidominant coroots. For β∈X∨β∈X∨beta inX^(vv)\beta \in \mathbb{X}^{\vee}β∈X∨, we set GrG(β)⊂GrGGrGâ¡(β)⊂GrGGr_(G)(beta)subGr_(G)\operatorname{Gr}_{G}(\beta) \subset \operatorname{Gr}_{G}GrGâ¡(β)⊂GrG as the corresponding I-orbit and set GrG(β):=∘¯GrG(β)⊂GrGGrGâ¡(β):=∘¯GrG(β)⊂GrGGr_(G)(beta):= bar(@)_(Gr_(G))(beta)subGr_(G)\operatorname{Gr}_{G}(\beta):=\bar{\circ}_{\mathrm{Gr}_{G}}(\beta) \subset \operatorname{Gr}_{G}GrGâ¡(β):=∘¯GrG(β)⊂GrG. We normalize so that GrG(β)GrGâ¡(β)Gr_(G)(beta)\operatorname{Gr}_{G}(\beta)GrGâ¡(β) is GGGGG-stable when β∈X<∨β∈X<∨beta inX_( < )^(vv)\beta \in \mathbb{X}_{<}^{\vee}β∈X<∨, and we have dimGrG(β)=−2|β|dimâ¡GrGâ¡(β)=−2|β|dim Gr_(G)(beta)=-2|beta|\operatorname{dim} \operatorname{Gr}_{G}(\beta)=-2|\beta|dimâ¡GrGâ¡(β)=−2|β| in such a case, where |β|:=∑i=1rβ(ϖi)|β|:=∑i=1r βϖi|beta|:=sum_(i=1)^(r)beta(Ï–_(i))|\beta|:=\sum_{i=1}^{r} \beta\left(\varpi_{i}\right)|β|:=∑i=1rβ(Ï–i).
We define
KT(GrG):=⋃β∈X∨KT(GrG(β)) and KG(GrG):=⋃β∈X∨KG(GrG(β))KTGrG:=⋃β∈X∨ KTGrGâ¡(β) and KGGrG:=⋃β∈X∨ KGGrGâ¡(β)K_(T)(Gr_(G)):=uuu_(beta inX^(vv))K_(T)(Gr_(G)(beta))quad" and "quadK_(G)(Gr_(G)):=uuu_(beta inXvv)K_(G)(Gr_(G)(beta))K_{T}\left(\operatorname{Gr}_{G}\right):=\bigcup_{\beta \in \mathbb{X}^{\vee}} K_{T}\left(\operatorname{Gr}_{G}(\beta)\right) \quad \text { and } \quad K_{G}\left(\operatorname{Gr}_{G}\right):=\bigcup_{\beta \in \mathbb{X} \vee} K_{G}\left(\operatorname{Gr}_{G}(\beta)\right)KT(GrG):=⋃β∈X∨KT(GrGâ¡(β)) and KG(GrG):=⋃β∈X∨KG(GrGâ¡(β))
These spaces are equipped with the convolution product, defined by the diagram
GrG×GrG←pG((z))×GrG→qG((z))××IGrG→ mult GrGGrG×GrGâ†pG((z))×GrG→qG((z))××IGrG→ mult GrGGr_(G)xxGr_(G)larr^(p)G((z))xxGr_(G)rarr"q"G((z))xxxx_(I)Gr_(G)rarr"" mult ""Gr_(G)\mathrm{Gr}_{G} \times \mathrm{Gr}_{G} \stackrel{p}{\leftarrow} G((z)) \times \operatorname{Gr}_{G} \xrightarrow{q} G((z)) \times \times_{\mathbf{I}} \mathrm{Gr}_{G} \xrightarrow{\text { mult }} \mathrm{Gr}_{G}GrG×GrGâ†pG((z))×GrG→qG((z))××IGrG→ mult GrG
as follows: For all cycles a,b∈KT(GrG)≅KI(GrG)a,b∈KTGrG≅KIGrGa,b inK_(T)(Gr_(G))~=K_(I)(Gr_(G))a, b \in K_{T}\left(\mathrm{Gr}_{G}\right) \cong K_{\mathrm{I}}\left(\mathrm{Gr}_{G}\right)a,b∈KT(GrG)≅KI(GrG), we find a left III\mathbf{I}I-equivariant class (a,b)(a,b)(a,b)(a, b)(a,b) on G((z))×IGrGG((z))×IGrGG((z))xx_(I)Gr_(G)G((z)) \times_{\mathrm{I}} \mathrm{Gr}_{G}G((z))×IGrG such that
This yields an associative product structure on KT(GrG)KTGrGK_(T)(Gr_(G))K_{T}\left(\mathrm{Gr}_{G}\right)KT(GrG) that contains a zero divisor. If we restrict ourselves to KG(GrG)KGGrGK_(G)(Gr_(G))K_{G}\left(\operatorname{Gr}_{G}\right)KG(GrG), then the algebra structure given by ⊙′⊙′o.^(')\odot^{\prime}⊙′ becomes commutative and integrally closed. Using an isomorphism KT(pt)⊗KG(pt)KG(GrG)≅KT(GrG)KT(pt)⊗KG(pt)KGGrG≅KTGrGK_(T)(pt)ox_(K_(G)(pt))K_(G)(Gr_(G))~=K_(T)(Gr_(G))K_{T}(\mathrm{pt}) \otimes_{K_{G}(\mathrm{pt})} K_{G}\left(\mathrm{Gr}_{G}\right) \cong K_{T}\left(\mathrm{Gr}_{G}\right)KT(pt)⊗KG(pt)KG(GrG)≅KT(GrG) of KT(pt)KT(pt)K_(T)(pt)K_{T}(\mathrm{pt})KT(pt)-modules, we find a multiplication ⊙⊙o.\odot⊙ of KT(GrG)KTGrGK_(T)(Gr_(G))K_{T}\left(\mathrm{Gr}_{G}\right)KT(GrG) that extends ⊙′⊙′o.^(')\odot^{\prime}⊙′ on KG(GrG)KGGrGK_(G)(Gr_(G))K_{G}\left(\mathrm{Gr}_{G}\right)KG(GrG) as a KT(pt)KT(pt)K_(T)(pt)K_{T}(\mathrm{pt})KT(pt)-algebra. This product ⊙⊙o.\odot⊙ coincides with a KKKKK-theoretic analogue of the Pontrjagin product (by the calculations in [47,$2.2][47,$2.2][47,$2.2][47, \$ 2.2][47,$2.2] ). In addition, we have
[OGrG(β+γ)]=[OGrG(β)]⊙[OGrG(γ)] for β,γ∈X<∨OGrGâ¡(β+γ)=OGrGâ¡(β)⊙OGrGâ¡(γ) for β,γ∈X<∨[O_(Gr_(G)(beta+gamma))]=[O_(Gr_(G)(beta))]o.[O_(Gr_(G)(gamma))]quad" for "quad beta,gamma inX_( < )^(vv)\left[\mathcal{O}_{\operatorname{Gr}_{G}(\beta+\gamma)}\right]=\left[\mathcal{O}_{\operatorname{Gr}_{G}(\beta)}\right] \odot\left[\mathcal{O}_{\operatorname{Gr}_{G}(\gamma)}\right] \quad \text { for } \quad \beta, \gamma \in \mathbb{X}_{<}^{\vee}[OGrGâ¡(β+γ)]=[OGrGâ¡(β)]⊙[OGrGâ¡(γ)] for β,γ∈X<∨
This yields a multiplicative system in KT(GrG)KTGrGK_(T)(Gr_(G))K_{T}\left(\mathrm{Gr}_{G}\right)KT(GrG), whose localization is denoted by KT(GrG)locKTGrGlocK_(T)(Gr_(G))_(loc)K_{T}\left(\operatorname{Gr}_{G}\right)_{\mathrm{loc}}KT(GrG)loc.
The (localized) small TTTTT-equivariant quantum KKKKK-group of BBB\mathscr{B}B is defined as a vector space
We denote the variable corresponding to β∈X∨β∈X∨beta inX^(vv)\beta \in \mathbb{X}^{\vee}β∈X∨ as QβQβQ^(beta)Q^{\beta}Qβ. The quantum KKKKK-theoretic product ⋆⋆***\star⋆ is a binary operation on qKT(B)locqKT(B)locqK_(T)(B)_(loc)q K_{T}(\mathscr{B})_{\mathrm{loc}}qKT(B)loc, defined by Givental [33] and Lee [61], whose value (a priori) belongs to a completion of qKT(B)loc qKT(B)loc qK_(T)(B)_("loc ")q K_{T}(\mathscr{B})_{\text {loc }}qKT(B)loc . It is one of the consequence of our analysis that ⋆⋆***\star⋆ preserves qKT(B)loc qKT(B)loc qK_(T)(B)_("loc ")q K_{T}(\mathscr{B})_{\text {loc }}qKT(B)loc . This is usually referred to as the finiteness of the quantum KKKKK-theoretic product (for BBB\mathscr{B}B ) in the literature [1,13], and is one of the most fundamental questions in the study of qKT(B)qKT(B)qK_(T)(B)q K_{T}(\mathscr{B})qKT(B). Lam-Li-Mihalcea-Shimozono [58] conjectured that:
Theorem 8.1 ([47]). We have an isomorphism of commutative algebras
for β,γ∈X<∨β,γ∈X<∨beta,gamma inX_( < )^(vv)\beta, \gamma \in \mathbb{X}_{<}^{\vee}β,γ∈X<∨ such that β(ϖi)<0βϖi<0beta(Ï–_(i)) < 0\beta\left(\varpi_{i}\right)<0β(Ï–i)<0 for every 1≤i≤r1≤i≤r1 <= i <= r1 \leq i \leq r1≤i≤r.
Note that a presentation of the ring qKT(B)qKT(B)qK_(T)(B)q K_{T}(\mathscr{B})qKT(B) for G=SL(n)G=SLâ¡(n)G=SL(n)G=\operatorname{SL}(n)G=SLâ¡(n) can be read-off from Givental-Lee [34], and a presentation of the ring KT(GrG)KTGrGK_(T)(Gr_(G))K_{T}\left(\mathrm{Gr}_{G}\right)KT(GrG) is obtained in BezrukavnikovFinkelberg-Mirković [6]. However, these are not enough to yield Theorem 8.1 (for G=SL(n))G=SL(n))G=SL(n))G=\mathrm{SL}(n))G=SL(n)) as the correspondence between Schubert bases is unclear.
We have an action of the nilpotent version Hnil Hnil H^("nil ")\mathscr{H}^{\text {nil }}Hnil of the double affine Hecke algebra (associated to G)G)G)G)G) on KT(GrG)KTGrGK_(T)(Gr_(G))K_{T}\left(\mathrm{Gr}_{G}\right)KT(GrG), coming from Kostant-Kumar [55]. In [47], we defined the TTTTT-equivariant KKKKK-group KT(QGrat )KTQGrat K_(T)(Q_(G)^("rat "))K_{T}\left(\mathbf{Q}_{G}^{\text {rat }}\right)KT(QGrat ) of QGrat QGrat Q_(G)^("rat ")\mathbf{Q}_{G}^{\text {rat }}QGrat based on the construction of the (T×Gmrot )T×Gmrot (T xxG_(m)^("rot "))\left(T \times \mathbb{G}_{m}^{\text {rot }}\right)(T×Gmrot ) equivariant KKKKK-group of QGrat QGrat Q_(G)^("rat ")\mathbf{Q}_{G}^{\text {rat }}QGrat in [52]. The III\mathbf{I}I-action on QGrat QGrat Q_(G)^("rat ")\mathbf{Q}_{G}^{\text {rat }}QGrat induces a HHnil HHnil HH^("nil ")\mathscr{H H}^{\text {nil }}HHnil -action on KT(QGrat )KTQGrat K_(T)(Q_(G)^("rat "))K_{T}\left(\mathbf{Q}_{G}^{\text {rat }}\right)KT(QGrat ).
The object KT(QGrat )KTQGrat K_(T)(Q_(G)^("rat "))K_{T}\left(\mathbf{Q}_{G}^{\text {rat }}\right)KT(QGrat ) needs a completion in order to admit an action of the line bundle twists by OQGrat (λ)(λ∈X)OQGrat (λ)(λ∈X)O_(Q_(G)^("rat "))(lambda)(lambda inX)\mathcal{O}_{\mathbf{Q}_{G}^{\text {rat }}}(\lambda)(\lambda \in \mathbb{X})OQGrat (λ)(λ∈X). It reflects the fact that the right-hand side of Theorem 5.3 (i.e., a global Weyl module) is infinite-dimensional in general, and hence the effect of ⊗OQGrat (ϖi)(1≤i≤r)⊗OQGrat ϖi(1≤i≤r)oxO_(Q_(G)^("rat "))(Ï–_(i))(1 <= i <= r)\otimes \mathcal{O}_{\mathbf{Q}_{G}^{\text {rat }}}\left(\varpi_{i}\right)(1 \leq i \leq r)⊗OQGrat (Ï–i)(1≤i≤r) requires infinitely many terms to describe.
Our main idea in the proof of Theorem 8.1 is to put QGrat QGrat Q_(G)^("rat ")\mathbf{Q}_{G}^{\text {rat }}QGrat into the picture:
Theorem 8.2 ([47, THEOREM c]). We have a commutative diagram
that respects the Schubert bases in each object. In addition, the map ΨΨPsi\PsiΨ is an embedding of representations of HHnil HHnil HH^("nil ")\mathscr{H} \mathscr{H}^{\text {nil }}HHnil , and the map ΨΨPsi\PsiΨ intertwines the tensor product with OQGrat (−ϖi)OQGrat −ϖiO_(Q_(G)^("rat "))(-Ï–_(i))\mathcal{O}_{\mathbf{Q}_{G}^{\text {rat }}}\left(-\varpi_{i}\right)OQGrat (−ϖi) in KT(QGrat )KTQGrat K_(T)(Q_(G)^("rat "))K_{T}\left(\mathbf{Q}_{G}^{\text {rat }}\right)KT(QGrat ) and the quantum product of OB(−ϖi)OB−ϖiO_(B)(-Ï–_(i))\mathcal{O}_{\mathcal{B}}\left(-\varpi_{i}\right)OB(−ϖi) on qKT(B)locqKT(B)locqK_(T)(B)_(loc)q K_{T}(\mathscr{B})_{\mathrm{loc}}qKT(B)loc for each 1≤i≤r1≤i≤r1 <= i <= r1 \leq i \leq r1≤i≤r.
The completion of KT(QGrat )KTQGrat K_(T)(Q_(G)^("rat "))K_{T}\left(\mathbf{Q}_{G}^{\text {rat }}\right)KT(QGrat ) is compatible with the standard completion of qKT(B)qKT(B)qK_(T)(B)q K_{T}(\mathscr{B})qKT(B) via the map ΨΨPsi\PsiΨ. Theorem 8.2 implies that the inverse of the operation ⋆OB(−ϖi)⋆OB−ϖi***O_(B)(-Ï–_(i))\star \mathcal{O}_{\mathcal{B}}\left(-\varpi_{i}\right)⋆OB(−ϖi) makes sense only after the completion of qKT(B)locqKT(B)locqK_(T)(B)_(loc)q K_{T}(\mathscr{B})_{\mathrm{loc}}qKT(B)loc.
Since the quantum KKKKK-theoretic correlators (see [33,61][33,61][33,61][33,61][33,61] ) satisfy neither the dimension axiom nor divisor axiom as in the theory of quantum cohomology, the proof of Theorem 8.2
must be necessarily different from Corollary 7.3. Our construction of the map ΨΨPsi\PsiΨ is based on the following two observations:
an interpretation of the ( GmGmG_(m)\mathbb{G}_{m}Gm-equivariant) quantum KKKKK-theoretic correlator
for each w∈W,λ∈X+w∈W,λ∈X+w in W,lambda inX_(+)w \in W, \lambda \in \mathbb{X}_{+}w∈W,λ∈X+as an element of KT(QGrat )KTQGrat K_(T)(Q_(G)^("rat "))K_{T}\left(\mathbf{Q}_{G}^{\text {rat }}\right)KT(QGrat ).
Here we can further interpret χ(EBβ(w,w0),πβ,w,w0∗OQ(w,w0tβ)(λ))χEBβw,w0,πβ,w,w0∗OQw,w0tβ(λ)chi(E_(B_(beta))(w,w_(0)),pi_(beta,w,w_(0))^(**)O_(Q(w,w_(0)t_(beta)))(lambda))\chi\left(\mathscr{E}_{\mathscr{B}_{\beta}}\left(w, w_{0}\right), \pi_{\beta, w, w_{0}}^{*} \mathcal{O}_{\mathscr{Q}\left(w, w_{0} t_{\beta}\right)}(\lambda)\right)χ(EBβ(w,w0),πβ,w,w0∗OQ(w,w0tβ)(λ)) using the shift operators of line bundles in quantum KKKKK-theory [35, PROPOSITION 2.13], and hence we obtain an (abstract) presentation of qKT(B)qKT(B)qK_(T)(B)q K_{T}(\mathscr{B})qKT(B) from (8.1) by the reconstruction theorem [35, PROPOSITION 2.12]. The identity (8.1) is a consequence of Theorem 7.4, and (8.2) is a consequence of compatible Frobenius splitting properties of QG(w,v)sQG(w,v)sQ_(G)(w,v)s\mathcal{Q}_{G}(w, v) \mathrm{s}QG(w,v)s and QGrat QGrat Q_(G)^("rat ")\mathbf{Q}_{G}^{\text {rat }}QGrat in char k>2k>2k > 2\mathbb{k}>2k>2 (see the explanation about the proof of Theorem 7.1).
There is a noncommutative version of Theorem 8.2, meaning that we include Gmrot Gmrot G_(m)^("rot ")\mathbb{G}_{m}^{\text {rot }}Gmrot (the variable " qqqqq " above) in each item [49].
9. FUNCTORIALITY OF QUANTUM KKKKK-GROUPS
We continue to work in the setting as in the previous section. In [50], we have presented analogues of Theorems 5.2, 5.3, and 7.1 for partial flag manifolds of GGGGG. Let us find a standard parabolic subgroup B⊂P⊂GB⊂P⊂GB sub P sub GB \subset P \subset GB⊂P⊂G and consider BP:=G/PBP:=G/PB_(P):=G//P\mathscr{B}_{P}:=G / PBP:=G/P. Our parabolic version of the semi-infinite flag manifold QG,Prat QG,Prat Q_(G,P)^("rat ")\mathbf{Q}_{G, P}^{\text {rat }}QG,Prat has its set of kkk\mathbb{k}k-valued points G((z))/(T⋅[P,P]((z)))G((z))/(Tâ‹…[P,P]((z)))G((z))//(T*[P,P]((z)))G((z)) /(T \cdot[P, P]((z)))G((z))/(Tâ‹…[P,P]((z))). The fiber of the natural map
is isomorphic to the semi-infinite flag manifold of [L,L][L,L][L,L][L, L][L,L], where L⊂PL⊂PL sub PL \subset PL⊂P is the maximal semisimple subgroup of PPPPP that contains TTTTT (the standard Levi subgroup). We also have the higher cohomology vanishing of equivariant line bundles on QG,Prat QG,Prat Q_(G,P)^("rat ")\mathbf{Q}_{G, P}^{\text {rat }}QG,Prat (or rather πP(QG))Ï€PQG{:pi_(P)(Q_(G)))\left.\pi_{P}\left(\mathbf{Q}_{G}\right)\right)Ï€P(QG)) as in Theorem 5.3. These are enough to yield a morphism
that intertwines appropriate line bundle twists (and analogous quantum multiplications). This yields a diagram
where we set QG,P:=πP(QG)QG,P:=Ï€PQGQ_(G,P):=pi_(P)(Q_(G))\mathbf{Q}_{G, P}:=\pi_{P}\left(\mathbf{Q}_{G}\right)QG,P:=Ï€P(QG).
The resulting map qKT(B)→qKT(BP)qKT(B)→qKTBPqK_(T)(B)rarr qK_(T)(B_(P))q K_{T}(\mathscr{B}) \rightarrow q K_{T}\left(\mathscr{B}_{P}\right)qKT(B)→qKT(BP) is, in fact, an algebra map [48], and is easy to describe. Note that we cannot have an analogous map between ordinary KKKKK-groups because of the higher direct images. It turns out this map sends Qαi∨Qαi∨Q^(alpha_(i)^(vv))Q^{\alpha_{i}^{\vee}}Qαi∨ to 1 for a simple coroot αi∨αi∨alpha_(i)^(vv)\alpha_{i}^{\vee}αi∨ belonging to LLLLL, and hence is not compatible with a naive generalization of the corresponding map in the Peterson isomorphism in homology [59].
Compared with the theory of flag manifolds, many precise results and constructions for QGrat QGrat Q_(G)^("rat ")\mathbf{Q}_{G}^{\text {rat }}QGrat are still missing. The most accessible set of problems might be to spell out analogues of numerous explicit formulas in classical Schubert calculus purely combinatorially by admitting geometric conclusions from [3,45,47,48,52][3,45,47,48,52][3,45,47,48,52][3,45,47,48,52][3,45,47,48,52] partly explained in the previous two sections. We close this note by briefly discussing some of other problems.
10.1. Categorifications of the coordinate rings
The homogeneous coordinate rings of Schubert varieties of a usual flag manifold, that are BBBBB-stable quotient rings of (2.1), can be seen as the Grothendieck groups of suitable categories equipped with cluster structures ([60]; see also Section 3.1). Hence, it is natural to expect categorifications of the homogeneous coordinate rings of QGrat (w)QGrat (w)Q_(G)^("rat ")(w)\mathbf{Q}_{G}^{\text {rat }}(w)QGrat (w) and BCthick BCthick B_(C)^("thick ")\mathscr{B}_{C}^{\text {thick }}BCthick . See also [21] and [43] for related problems and partial answers.
10.2. Peterson isomorphism in quantum cohomology
The Peterson isomorphism in quantum cohomology [59,74] is an analogue of Theorem 8.1 for homology. We may apply Corollary 7.3 to [69] (that is an essential ingredient in [59]) to utilize QGrat QGrat Q_(G)^("rat ")\mathbf{Q}_{G}^{\text {rat }}QGrat in its proof (that looks similar to the original strategy in [74]). However, we do not know an analogue of Theorem 8.2 as we lack a proper definition of H∙(QG)H∙QGH^(∙)(Q_(G))H^{\bullet}\left(\mathbf{Q}_{G}\right)H∙(QG).
10.3. Constructible sheaves on semi-infinite flags
In representation-theoretic analysis on BBB\mathscr{B}B, we sometimes encounter constructible sheaves that are not NNNNN-equivariant. Also, we want some notion of (co)homology of QGQGQ_(G)\mathbf{Q}_{G}QG in
Section 10.2. Therefore, it is desirable to understand constructible sheaves on QGQGQ_(G)\mathbf{Q}_{G}QG following [7]. The resulting objects should have connection to [30]. Note that the combinatorics that should be satisfied by the I-equivariant sheaves (equipped with Frobenius endomorphisms) have been worked out in detail [62,65][62,65][62,65][62,65][62,65].
10.4. Tensor product decompositions
The tensor product decomposition of rational representations of GGGGG is deeply connected with our whole story due to the presentation (2.1). In [57], the geometry of flag varieties is used to deduce subtle information on the tensor products beyond the classical Littlewood-Richardson rule. It would be interesting to pursue their analogues in QGrat QGrat Q_(G)^("rat ")\mathbf{Q}_{G}^{\text {rat }}QGrat , possibly utilizing some modular interpretation [11] and connecting with the perspectives in [5].
10.5. The cotangent bundle of semi-infinite flags
A version of the cotangent bundle of QGrat QGrat Q_(G)^("rat ")\mathbf{Q}_{G}^{\text {rat }}QGrat would make it possible to compare our results with the perspectives in [21,67,73]. In addition, its quantization should realize some numerics in Section 10.3. The author hopes to say a bit more on this in St. Petersburg.
ACKNOWLEDGMENTS
The works presented here could not be carried out without suggestions and interest by Misha Finkelberg. The author would like to express his deepest gratitude to him. The author also would like to thank Ivan Cherednik and Thomas Lam for sharing their insights over years, and Noriyuki Abe and Toshiyuki Tanisaki for their comments.
FUNDING
This work was partially supported by JSPS KAKENHI Grant Numbers JP26287004, JP19H01782, and JPJSBP120213210.
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SYU KATO
Department of Mathematics, Kyoto University, Oiwake Kita-Shirakawa, Sakyo, Kyoto 6068502, Japan, syuchan@math.kyoto-u.ac.jp
CHARACTER ESTIMATES FOR FINITE SIMPLE GROUPS AND APPLICATIONS
MICHAEL J. LARSEN
ABSTRACT
Let GGGGG be a finite simple group, χχchi\chiχ an irreducible complex character, and ggggg an element of GGGGG. It is often desirable to have upper bounds for |χ(g)||χ(g)||chi(g)||\chi(g)||χ(g)| in terms of χ(1)χ(1)chi(1)\chi(1)χ(1) and some measure of the regularity of ggggg. This paper reviews what is known in this direction and presents typical applications of such bounds: to proving certain products of conjugacy classes cover GGGGG, to solving word equations over GGGGG, and to counting homomorphisms from a Fuchsian group to GGGGG.
Let GGGGG be a finite group, χχchi\chiχ the character of an irreducible complex representation ρÏrho\rhoÏ of GGGGG, and ggggg an element of GGGGG. As the eigenvalues of ρ(g)Ï(g)rho(g)\rho(g)Ï(g) are roots of unity, the bound |χ(g)|≤χ(1)|χ(g)|≤χ(1)|chi(g)| <= chi(1)|\chi(g)| \leq \chi(1)|χ(g)|≤χ(1) is trivial. For central elements ggggg, no stronger upper bound than χ(1)χ(1)chi(1)\chi(1)χ(1) is possible. However, according to Schur, we know that
∑g∈Gχ(g)χ(g)¯=|G|∑g∈G χ(g)χ(g)¯=|G|sum_(g in G)chi(g) bar(chi(g))=|G|\sum_{g \in G} \chi(g) \overline{\chi(g)}=|G|∑g∈Gχ(g)χ(g)¯=|G|
and since χ(x)=χ(g)χ(x)=χ(g)chi(x)=chi(g)\chi(x)=\chi(g)χ(x)=χ(g) for all xxxxx in the conjugacy class gGgGg^(G)g^{G}gG, we obtain the centralizer bound
Other known upper bounds typically hold only for special classes of groups.
This paper reviews what is known about character bounds when GGGGG is a finite simple group or is closely related to such a group. There is a substantial literature on upper bounds for character ratios |χ(g)|χ(1)|χ(g)|χ(1)(|chi(g)|)/(chi(1))\frac{|\chi(g)|}{\chi(1)}|χ(g)|χ(1); see Martin Liebeck's survey [29] for recent results and applications in the case of groups of Lie type. These bounds are typically weakest for characters χχchi\chiχ of low degree, which points to the desirability of exponential bounds, that is, bounds of the form |χ(g)|≤χ(1)α(g)|χ(g)|≤χ(1)α(g)|chi(g)| <= chi(1)^(alpha(g))|\chi(g)| \leq \chi(1)^{\alpha(g)}|χ(g)|≤χ(1)α(g), where the size of α(g)α(g)alpha(g)\alpha(g)α(g) is typically related to the size of the centralizer of ggggg compared to |G||G||G||G||G|. The next two sections focus on alternating groups and groups of Lie type, respectively. The remaining sections give some applications of these results and present some open problems.
2. SYMMETRIC AND ALTERNATING GROUPS
Motivated by questions in probability theory, a number of people have considered character ratio bounds for symmetric groups. In this series of groups, unlike groups of Lie type, character ratios for nontrivial elements and nontrivial characters can be arbitrarily close to 1 . The worst case for G=SnG=SnG=S_(n)G=\mathrm{S}_{n}G=Sn is the ratio n−3n−1n−3n−1(n-3)/(n-1)\frac{n-3}{n-1}n−3n−1, achieved when ggggg is a transposition and χχchi\chiχ is a character of degree n−1n−1n-1n-1n−1. Persi Diaconis and Mehrdad Shahshahani considered the case that ggggg is a transposition and χχchi\chiχ is any irreducible character, proving in [4] that if both the first row and the first column of the Young diagram for χ=χλχ=χλchi=chi_(lambda)\chi=\chi_{\lambda}χ=χλ have length ≤n/2≤n/2<= n//2\leq n / 2≤n/2, then the character ratio is less than 1/21/21//21 / 21/2, while if, for instance, the first row satisfies λ1>n/2λ1>n/2lambda_(1) > n//2\lambda_{1}>n / 2λ1>n/2, then
A similar bound was given by Leopold Flatto, Andrew Odlyzko, and David Wales [8, тHEoREM 5.2].
Yuval Roichman [39] gave a character bound of the form
|χ(g)|χ(1)≤max(λ1/n,λ1′/n,c)supp(g)|χ(g)|χ(1)≤maxλ1/n,λ1′/n,csuppâ¡(g)(|chi(g)|)/(chi(1)) <= max(lambda_(1)//n,lambda_(1)^(')//n,c)^(supp(g))\frac{|\chi(g)|}{\chi(1)} \leq \max \left(\lambda_{1} / n, \lambda_{1}^{\prime} / n, c\right)^{\operatorname{supp}(g)}|χ(g)|χ(1)≤max(λ1/n,λ1′/n,c)suppâ¡(g)
where supp(g)suppâ¡(g)supp(g)\operatorname{supp}(g)suppâ¡(g) denotes the number of elements of {1,…,n}{1,…,n}{1,dots,n}\{1, \ldots, n\}{1,…,n} not fixed by ggggg, and c<1c<1c < 1c<1c<1 is an absolute constant. This reflects the fact that elements with high support tend to have
small centralizers. The bound is quite good when χχchi\chiχ has small degree. However, for large nnnnn, most characters of SnSnS_(n)\mathrm{S}_{n}Sn have degree greater than AnAnA^(n)A^{n}An for any fixed AAAAA, and for such characters, Roichman's bound is weaker than the centralizer bound for most elements g∈Gg∈Gg in Gg \in Gg∈G.
Philippe Biane [3] gave character ratio bounds for elements of bounded support and "balanced" characters, namely those where λ1/nλ1/nlambda_(1)//sqrtn\lambda_{1} / \sqrt{n}λ1/n and λ1′/nλ1′/nlambda_(1)^(')//sqrtn\lambda_{1}^{\prime} / \sqrt{n}λ1′/n are bounded. By the work of Logan-Shepp [34] and VerÅ¡ik-Kerov [44], high degree characters are typically balanced. To be more precise, this is true for characters chosen randomly, weighted by the Plancherel measure. Amarpreet Rattan and Piotr Åšniady [38] generalized Biane's character bound so it applies whenever supp(g)suppâ¡(g)supp(g)\operatorname{supp}(g)suppâ¡(g) is small enough compared to nnnnn; if ggggg cannot be expressed as the product of less than πÏ€pi\piÏ€ transpositions, then
which simultaneously improves on the results of [39] and [38].
Thomas Müller and Jan-Christoph Schlage-Puchta gave a character bound of exponential type [37, THEOREM B] which is good in a wide variety of situations. They proved that |χ(g)|≤χ(1)α(g)|χ(g)|≤χ(1)α(g)|chi(g)| <= chi(1)^(alpha(g))|\chi(g)| \leq \chi(1)^{\alpha(g)}|χ(g)|≤χ(1)α(g), where
Being exponential, it works well whether χ(1)χ(1)chi(1)\chi(1)χ(1) is large or small. The exponent is optimal, up to a multiplicative constant, for elements ggggg consisting of many cycles, for instance, for involutions. However, it can be greatly improved upon for elements consisting of few cycles. In particular, α(g)α(g)alpha(g)\alpha(g)α(g) is no smaller when ggggg is an nnnnn-cycle than when it is of shape 2n/22n/22^(n//2)2^{n / 2}2n/2.
Sergey Fomin and Nathan Lulov [9] gave a bound specifically for elements ggggg of the shape rn/rrn/rr^(n//r)r^{n / r}rn/r. For fixed rrrrr and varying nnnnn, it takes the form
so it is essentially a bound of exponential type. Aner Shalev and I gave an exponential bound [22] for elements ggggg of arbitrary shape 1a12a2…1a12a2…1^(a_(1))2^(a_(2))dots1^{a_{1}} 2^{a_{2}} \ldots1a12a2… which is roughly comparable in strength to the Fomin-Lulov bound. Define the sequence e1,e2,…e1,e2,…e_(1),e_(2),dotse_{1}, e_{2}, \ldotse1,e2,… such that for all k≥1k≥1k >= 1k \geq 1k≥1,
ne1+⋯+ek=∑i=1kiaine1+⋯+ek=∑i=1k iain^(e_(1)+cdots+e_(k))=sum_(i=1)^(k)ia_(i)n^{e_{1}+\cdots+e_{k}}=\sum_{i=1}^{k} i a_{i}ne1+⋯+ek=∑i=1kiai
This result is stronger than the exponential bound of Müller-Schlage-Puchta for almost all elements but inferior to it when the number of fixed points of ggggg is very large.
None of these bounds can compete with the centralizer bound for elements consisting of very few cycles, for instance, for nnnnn-cycles, where the centralizer bound gives |χ(g)|≤n|χ(g)|≤n|chi(g)| <= sqrtn|\chi(g)| \leq \sqrt{n}|χ(g)|≤n. For such elements, the Murnaghan-Nakayama rule asserts |χ(g)|≤1|χ(g)|≤1|chi(g)| <= 1|\chi(g)| \leq 1|χ(g)|≤1, which is obviously optimal.
From symmetric group bounds, we easily obtain alternating group bounds of comparable strength. Recall that for λ≠λ′λ≠λ′lambda!=lambda^(')\lambda \neq \lambda^{\prime}λ≠λ′, the characters χλχλchi_(lambda)\chi_{\lambda}χλ and χλ′χλ′chi_(lambda^('))\chi_{\lambda^{\prime}}χλ′ restrict to the same irreducible character of AnAnA_(n)\mathrm{A}_{n}An. All other irreducible characters of AnAnA_(n)\mathrm{A}_{n}An arise from partitions satisfying λ=λ′λ=λ′lambda=lambda^(')\lambda=\lambda^{\prime}λ=λ′; for each such λλlambda\lambdaλ, the restriction of χλχλchi_(lambda)\chi_{\lambda}χλ to AnAnA_(n)\mathrm{A}_{n}An decomposes as a sum of two irreducibles χλ1χλ1chi_(lambda)^(1)\chi_{\lambda}^{1}χλ1 and χλ2χλ2chi_(lambda)^(2)\chi_{\lambda}^{2}χλ2. The χλiχλichi_(lambda)^(i)\chi_{\lambda}^{i}χλi take the character value χλ(g)/2χλ(g)/2chi_(lambda)(g)//2\chi_{\lambda}(g) / 2χλ(g)/2 for all g∈Sn∖Cg∈Sn∖Cg inS_(n)\\Cg \in \mathrm{S}_{n} \backslash Cg∈Sn∖C, where CCCCC is a single SnSnS_(n)\mathrm{S}_{n}Sn-conjugacy class which decomposes into two AnAnA_(n)\mathrm{A}_{n}An-conjugacy classes. For elements of CCCCC, a theorem of Frobenius gives character values, which are of the form
where ni=λi−ini=λi−in_(i)=lambda_(i)-in_{i}=\lambda_{i}-ini=λi−i for 1≤i≤k1≤i≤k1 <= i <= k1 \leq i \leq k1≤i≤k. Character degree estimates, like those in [22], now imply that |χλi(g)|≤χλ(1)εχλi(g)≤χλ(1)ε|chi_(lambda)^(i)(g)| <= chi_(lambda)(1)^(epsi)\left|\chi_{\lambda}^{i}(g)\right| \leq \chi_{\lambda}(1)^{\varepsilon}|χλi(g)|≤χλ(1)ε whenever nnnnn is sufficiently large compared to ε>0ε>0epsi > 0\varepsilon>0ε>0.
3. GROUPS OF LIE TYPE
Character estimates for finite simple groups of Lie type go back to the work of David Gluck [13-15]. Unlike in the case of alternating and symmetric groups, there is a uniform bound [15] on character ratios for nontrivial characters and nontrivial ggggg, namely
When the cardinality qqqqq of the field of definition of GGGGG is large, this upper bound can be improved; Gluck [14] gives an upper bound of the form C/qC/qC//sqrtqC / \sqrt{q}C/q for large qqqqq. The qqqqq-exponent is optimal, since for odd q,PSL2(q)q,PSL2â¡(q)q,PSL_(2)(q)q, \operatorname{PSL}_{2}(q)q,PSL2â¡(q) has characters of degree q+12q+12(q+1)/(2)\frac{q+1}{2}q+12 or q−12q−12(q-1)/(2)\frac{q-1}{2}q−12, and the value of such a character at a nontrivial unipotent element ggggg is ±1±(−1)q−12q2±1±(−1)q−12q2(+-1+-sqrt((-1)^((q-1)/(2))q))/(2)\frac{ \pm 1 \pm \sqrt{(-1)^{\frac{q-1}{2}} q}}{2}±1±(−1)q−12q2.
We cannot expect that character ratios go to 0 as the order of a classical group goes to infinity. For instance, let G=PSLr+1(q)G=PSLr+1(q)G=PSL_(r+1)(q)G=\mathrm{PSL}_{r+1}(q)G=PSLr+1(q). The permutation representation associated with the action of GGGGG on PFqrPFqrPF_(q)^(r)\mathbb{P} F_{q}^{r}PFqr can be expressed as χ+1χ+1chi+1\chi+1χ+1, for χχchi\chiχ irreducible. Let ggggg be the image of a transvection in SLr+1(Fq)SLr+1â¡FqSL_(r+1)(F_(q))\operatorname{SL}_{r+1}\left(\mathbb{F}_{q}\right)SLr+1â¡(Fq) in GGGGG. Then the fixed points of ggggg form a hyperplane in PPqnPPqnPP_(q)^(n)\mathbb{P P}_{q}^{n}PPqn, and χ(g)=qn−1+qn−2+⋯+qχ(g)=qn−1+qn−2+⋯+qchi(g)=q^(n-1)+q^(n-2)+cdots+q\chi(g)=q^{n-1}+q^{n-2}+\cdots+qχ(g)=qn−1+qn−2+⋯+q. Thus,
Defining the support supp(g)suppâ¡(g)supp(g)\operatorname{supp}(g)suppâ¡(g) as the smallest codimension of any eigenspace of ggggg for the natural projective representation of GGGGG, the elements ggggg in the above example have constant support 1 even as the rank of GGGGG goes to infinity. Shalev, Pham Huu Tiep, and I proved [24, THEOREM 4.3.6] that as the support goes to infinity, the character ratio goes to 0 :
This falls well short of a uniform exponential character bound, even for elements of maximal support. Robert Guralnick, Tiep, and I found uniform exponential bounds for elements ggggg whose centralizer is small compared to the order of GGGGG. For instance, we proved [16, THEOREM 1.4] that if GGGGG is of the form PSLn(q)PSLnâ¡(q)PSL_(n)(q)\operatorname{PSL}_{n}(q)PSLnâ¡(q) or PSUn(q)PSUnâ¡(q)PSU_(n)(q)\operatorname{PSU}_{n}(q)PSUnâ¡(q) and |CG(g)|≤qn2/12CG(g)≤qn2/12|C_(G)(g)| <= q^(n^(2)//12)\left|C_{G}(g)\right| \leq q^{n^{2} / 12}|CG(g)|≤qn2/12, then |χ(g)|≤χ(1)8/9|χ(g)|≤χ(1)8/9|chi(g)| <= chi(1)^(8//9)|\chi(g)| \leq \chi(1)^{8 / 9}|χ(g)|≤χ(1)8/9. More generally, but less explicitly, we proved [17, THEOREM 1.3] that for all ε>0ε>0epsi > 0\varepsilon>0ε>0, there exists δ>0δ>0delta > 0\delta>0δ>0 such that |CG(g)|≤|G|δCG(g)≤|G|δ|C_(G)(g)| <= |G|^(delta)\left|C_{G}(g)\right| \leq|G|^{\delta}|CG(g)|≤|G|δ implies |χ(g)|≤χ(1)ε|χ(g)|≤χ(1)ε|chi(g)| <= chi(1)^(epsi)|\chi(g)| \leq \chi(1)^{\varepsilon}|χ(g)|≤χ(1)ε. However, the method of these papers applies only to elements with small centralizer, for instance, it does not give any bound at all for involutions.
This defect was remedied in the sequel [28], which proved that for all positive δ<1δ<1delta < 1\delta<1δ<1 there exists ε<1ε<1epsi < 1\varepsilon<1ε<1 such that |CG(g)|≤|G|δCG(g)≤|G|δ|C_(G)(g)| <= |G|^(delta)\left|C_{G}(g)\right| \leq|G|^{\delta}|CG(g)|≤|G|δ implies |χ(g)|≤χ(1)ε|χ(g)|≤χ(1)ε|chi(g)| <= chi(1)^(epsi)|\chi(g)| \leq \chi(1)^{\varepsilon}|χ(g)|≤χ(1)ε. More precisely, |χ(g)|≤|χ(g)|≤|chi(g)| <=|\chi(g)| \leq|χ(g)|≤χ(1)α(g)χ(1)α(g)chi(1)^(alpha(g))\chi(1)^{\alpha(g)}χ(1)α(g) where
and c>0c>0c > 0c>0c>0 is an absolute constant, which can be made explicit (but is, unfortunately, extremely small). This theorem holds more generally for quasisimple finite groups of Lie type.
For many elements ggggg in a classical group of rank rrrrr, much better exponents are available, thanks to the work of Roman Bezrukavnikov, Liebeck, Shalev, and Tiep [2]. For qqqqq odd, if the centralizer of ggggg is a proper split Levi subgroup, then |χ(g)|≤f(r)χ(1)α(g)|χ(g)|≤f(r)χ(1)α(g)|chi(g)| <= f(r)chi(1)^(alpha(g))|\chi(g)| \leq f(r) \chi(1)^{\alpha(g)}|χ(g)|≤f(r)χ(1)α(g), where α(g)α(g)alpha(g)\alpha(g)α(g) is an explicitly computable rational number which is known to be optimal in many cases. This idea was further developed by Jay Taylor and Tiep, who proved [43], among other things, that for every nontrivial element g∈PSLn(q)g∈PSLnâ¡(q)g inPSL_(n)(q)g \in \operatorname{PSL}_{n}(q)g∈PSLnâ¡(q),
All of these estimates are poor for elements with small centralizers, such as regular elements. A general result, due to Shelly Garion, Alexander Lubotzky, and myself, which sometimes gives reasonably good bounds for regular elements, is the following [10, THEOREM 3]. Let GGGGG be a finite group, not necessarily simple, and ggggg an element of GGGGG whose centralizer AAAAA is abelian. Suppose A1,…,AnA1,…,AnA_(1),dots,A_(n)A_{1}, \ldots, A_{n}A1,…,An are subgroups of AAAAA not containing ggggg such that the centralizer of every element of A∖⋃iAiA∖⋃i AiA\\uuu_(i)A_(i)A \backslash \bigcup_{i} A_{i}A∖⋃iAi is AAAAA. Then, for every irreducible character of GGGGG,
For example, this gives an upper bound of 2(n−1)2/32(n−1)2/32(n-1)^(2)//sqrt32(n-1)^{2} / \sqrt{3}2(n−1)2/3 for |χ(g)||χ(g)||chi(g)||\chi(g)||χ(g)| when G=PSLn(q)G=PSLnâ¡(q)G=PSL_(n)(q)G=\operatorname{PSL}_{n}(q)G=PSLnâ¡(q) and ggggg is the image of an element with irreducible characteristic polynomial. It would be nice to have optimal upper bounds for |χ(g)||χ(g)||chi(g)||\chi(g)||χ(g)| for general regular semisimple elements ggggg.
4. PRODUCTS OF CONJUGACY CLASSES
If C1,…,CnC1,…,CnC_(1),dots,C_(n)C_{1}, \ldots, C_{n}C1,…,Cn are conjugacy classes of a finite group GGGGG, then the number NNNNN of nnnnn tuples (g1,…,gn)∈C1×⋯×Cng1,…,gn∈C1×⋯×Cn(g_(1),dots,g_(n))inC_(1)xx cdots xxC_(n)\left(g_{1}, \ldots, g_{n}\right) \in C_{1} \times \cdots \times C_{n}(g1,…,gn)∈C1×⋯×Cn satisfying g1g2⋯gn=1g1g2⋯gn=1g_(1)g_(2)cdotsg_(n)=1g_{1} g_{2} \cdots g_{n}=1g1g2⋯gn=1 is given by the Frobenius formula
where χχchi\chiχ ranges over all irreducible characters of GGGGG. In conjunction with upper bounds for the |χ(Ci)|χCi|chi(C_(i))|\left|\chi\left(C_{i}\right)\right||χ(Ci)|, this can sometimes be used to prove that N≠0N≠0N!=0N \neq 0N≠0, as the contribution from χ=1χ=1chi=1\chi=1χ=1 often dominates the sum. Exponential bounds for the χ(Ci)χCichi(C_(i))\chi\left(C_{i}\right)χ(Ci) are especially convenient, since results of Liebeck and Shalev [32] give a great deal of information about when we can expect
A well-known conjecture attributed to Thompson asserts that for every finite simple group GGGGG, there exists a conjugacy class CCCCC such that C2=GC2=GC^(2)=GC^{2}=GC2=G. Thanks to work of Erich Ellers and Nikolai Gordeev [6], we know that this is true except for a list of possible counterexamples, all finite simple groups of Lie type with q≤8q≤8q <= 8q \leq 8q≤8. Tiep and I used our uniform exponential bounds to show that several of the infinite families on this list, in particular, the symplectic groups for all q≥2q≥2q >= 2q \geq 2q≥2, can be eliminated in sufficiently high rank [28, Ñ‚HEoREM 7.7]. It would be interesting if these results could be extended to the remaining families on the list, giving an asymptotic version of Thompson's conjecture.
Andrew Gleason and Cheng-hao Xu [18,19] proved Thompson's conjecture for alternating groups, using the conjugacy class of an nnnnn-cycle if nnnnn is odd or a permutation of shape 21(n−2)121(n−2)12^(1)(n-2)^(1)2^{1}(n-2)^{1}21(n−2)1 if nnnnn is even. In [22, THEOREM 1.13], Shalev and I proved that in the limit n→∞n→∞n rarr oon \rightarrow \inftyn→∞ the probability that a randomly chosen g∈Ang∈Ang inA_(n)g \in \mathrm{A}_{n}g∈An belongs to a conjugacy class with C2=AnC2=AnC^(2)=A_(n)C^{2}=\mathrm{A}_{n}C2=An rapidly approaches 1 .
The analogous claim cannot be true for all finite simple groups since C2=GC2=GC^(2)=GC^{2}=GC2=G implies that C=C−1C=C−1C=C^(-1)C=C^{-1}C=C−1, and for, e.g., PSL3(q)PSL3â¡(q)PSL_(3)(q)\operatorname{PSL}_{3}(q)PSL3â¡(q) as q→∞q→∞q rarr ooq \rightarrow \inftyq→∞, the probability that a random element is real goes to 0 . However, there are several variants of this question which do not have an obvious counterexample. As the order of GGGGG tends to infinity, does the probability that a random real element belongs to a conjugacy class with C2=GC2=GC^(2)=GC^{2}=GC2=G approach 1? Does the probability that a random element ggggg belongs to a conjugacy class CCCCC with C2∪{1}=GC2∪{1}=GC^(2)uu{1}=GC^{2} \cup\{1\}=GC2∪{1}=G approach 1? Also, as the order of GGGGG tends to infinity, does the probability that a random element belongs to a conjugacy class with CC−1=GCC−1=GCC^(-1)=GC C^{-1}=GCC−1=G approach 1?
The weaker claim that every element g∈Gg∈Gg in Gg \in Gg∈G lies in CC−1CC−1CC^(-1)C C^{-1}CC−1 for some conjugacy class (depending, perhaps, on ggggg ) is equivalent to the statement that every element of GGGGG is a commutator. This was was an old conjecture of Ore and is now a theorem of Liebeck, Eamonn O'Brien, Shalev, and Tiep [30].
One can also ask about S2S2S^(2)S^{2}S2 where SSSSS is an arbitrary conjugation-invariant subset of GGGGG. On naive probabilistic grounds, it might seem plausible that given ε>0ε>0epsi > 0\varepsilon>0ε>0 fixed, for GGGGG sufficiently large, every normal subset of GGGGG with at least ε|G|ε|G|epsi|G|\varepsilon|G|ε|G| elements satisfies S2=GS2=GS^(2)=GS^{2}=GS2=G. However, a moment's reflection shows that, unless ε>12ε>12epsi > (1)/(2)\varepsilon>\frac{1}{2}ε>12, there is no reason to expect 1∈S21∈S21inS^(2)1 \in S^{2}1∈S2.
Is it true, for GGGGG sufficiently large, that S2∪{1}=GS2∪{1}=GS^(2)uu{1}=GS^{2} \cup\{1\}=GS2∪{1}=G ? For alternating groups and for groups of Lie type in bounded rank, the answer is affirmative [26], but we do not know in general.
In a different direction, given a conjugacy class CCCCC, how large must nnnnn be so that the nnnnnth power CnCnC^(n)C^{n}Cn is all of GGGGG ? More generally, given conjugacy classes C1,…,CnC1,…,CnC_(1),dots,C_(n)C_{1}, \ldots, C_{n}C1,…,Cn with sufficiently strong character bounds, the Frobenius formula can be used to show that each element of GGGGG is represented as a product g1⋯gng1⋯gng_(1)cdotsg_(n)g_{1} \cdots g_{n}g1⋯gn, with gi∈Cigi∈Cig_(i)inC_(i)g_{i} \in C_{i}gi∈Ci, in approximately |C1|⋯|Cn||G|C1⋯Cn|G|(|C_(1)|cdots|C_(n)|)/(|G|)\frac{\left|C_{1}\right| \cdots\left|C_{n}\right|}{|G|}|C1|⋯|Cn||G| ways. For instance, it follows from the exponential character bounds given above that there exists an absolute constant kkkkk such that if GGGGG is a finite simple group of Lie type and C1,…,CnC1,…,CnC_(1),dots,C_(n)C_{1}, \ldots, C_{n}C1,…,Cn are conjugacy classes in GGGGG satisfying |C1|⋯|Cn|>|G|kC1⋯Cn>|G|k|C_(1)|cdots|C_(n)| > |G|^(k)\left|C_{1}\right| \cdots\left|C_{n}\right|>|G|^{k}|C1|⋯|Cn|>|G|k, then for each g∈Gg∈Gg in Gg \in Gg∈G,
Via Lang-Weil estimates, this further implies that if C_1,…,C_nC_1,…,C_nC__(1),dots,C__(n)\underline{C}_{1}, \ldots, \underline{C}_{n}C_1,…,C_n are conjugacy classes of a simple algebraic group G_G_G_\underline{G}G_, and
dimC_1+⋯+dimC_n>kdimG_dimâ¡C_1+⋯+dimâ¡C_n>kdimâ¡G_dim C__(1)+cdots+dim C__(n) > k dim G_\operatorname{dim} \underline{C}_{1}+\cdots+\operatorname{dim} \underline{C}_{n}>k \operatorname{dim} \underline{G}dimâ¡C_1+⋯+dimâ¡C_n>kdimâ¡G_
then the product morphism of varieties C_1×⋯×C_n→G_C_1×⋯×C_n→G_C__(1)xx cdots xxC__(n)rarrG_\underline{C}_{1} \times \cdots \times \underline{C}_{n} \rightarrow \underline{G}C_1×⋯×C_n→G_ has the property that every fiber is of dimension dimC_1+⋯+dimC_n−dimG_dimâ¡C_1+⋯+dimâ¡C_n−dimâ¡G_dim C__(1)+cdots+dim C__(n)-dim G_\operatorname{dim} \underline{C}_{1}+\cdots+\operatorname{dim} \underline{C}_{n}-\operatorname{dim} \underline{G}dimâ¡C_1+⋯+dimâ¡C_n−dimâ¡G_.
In the special case that C1=⋯=Cn=CC1=⋯=Cn=CC_(1)=cdots=C_(n)=CC_{1}=\cdots=C_{n}=CC1=⋯=Cn=C, the question of the distribution of products g1⋯gn,gi∈Cg1⋯gn,gi∈Cg_(1)cdotsg_(n),g_(i)in Cg_{1} \cdots g_{n}, g_{i} \in Cg1⋯gn,gi∈C, can be expressed in terms of the mixing time of the random walk on the Cayley graph of ( G,CG,CG,CG, CG,C ). A consequence of the exponential character bounds [28] is that for groups of Lie type, the mixing time of such a random walk is O(log|G|/log|C|)O(logâ¡|G|/logâ¡|C|)O(log |G|//log |C|)O(\log |G| / \log |C|)O(logâ¡|G|/logâ¡|C|). This is the same order of growth as the diameter of the Cayley graph, thus settling conjectures of Lubotzky [35, P. 179] and Shalev [42, CONJECTURE 4.3].
The situation is different for alternating groups G=AnG=AnG=A_(n)G=\mathrm{A}_{n}G=An. For instance, if CCCCC is the class of 3-cycles and n≥6n≥6n >= 6n \geq 6n≥6, then log|G|/log|C|<nlogâ¡|G|/logâ¡|C|<nlog |G|//log |C| < n\log |G| / \log |C|<nlogâ¡|G|/logâ¡|C|<n, and C⌊n/2⌋=GC⌊n/2⌋=GC^(|__ n//2__|)=GC^{\lfloor n / 2\rfloor}=GC⌊n/2⌋=G [5, THEOREM 9.1]. However, for any fixed kkkkk, the probability that the product of knknknk nkn random 3 -cycles gigig_(i)g_{i}gi fixes 1 is at least the probability that each individual gigig_(i)g_{i}gi fixes 1 , which goes to e−3ke−3ke^(-3k)e^{-3 k}e−3k as n→∞n→∞n rarr oon \rightarrow \inftyn→∞. Thus the expected number of fixed points of g1⋯gng1⋯gng_(1)cdotsg_(n)g_{1} \cdots g_{n}g1⋯gn grows linearly with nnnnn. It would be interesting to know, for general C⊂AnC⊂AnC subA_(n)C \subset \mathrm{A}_{n}C⊂An, what the mixing time is.
5. WARING'S PROBLEM
Waring's problem for finite simple groups originally meant the following question. Does there exist a function f:N→Nf:N→Nf:NrarrNf: \mathbb{N} \rightarrow \mathbb{N}f:N→N such that for all positive integers nnnnn and all sufficiently large finite simple groups GGGGG (in terms of nnnnn ), every element of GGGGG is a product of f(n)nf(n)nf(n)nf(n) nf(n)nth powers? Positive solutions were given by Martinez-Zelmanov [36] and Saxl-Wilson [40].
This can be extended as follows. Let wwwww denote a nontrivial element in any free group FdFdF_(d)F_{d}Fd. For every finite simple group G,wG,wG,wG, wG,w determines a function Gd→GGd→GG^(d)rarr GG^{d} \rightarrow GGd→G. We replace the nnnnnth powers with word values, that is, elements of GGGGG in the image of wwwww. Liebeck and Shalev proved [31] that for GGGGG sufficiently large (in terms of wwwww ), every element of GGGGG can be written as a product of a bounded number of word values (where the bound may depend on wwwww, just as
in the classical version of Waring's problem, the minimum number of the nnnnnth powers needed to represent a given integer may depend on nnnnn ).
It was therefore, perhaps, surprising when Shalev proved [41] that the Waring number for finite simple groups is uniform in wwwww and is, in fact, at most three. This has now been improved to the optimal bound, two [23,24]. More generally, for any two nontrivial words w1w1w_(1)w_{1}w1 and w2w2w_(2)w_{2}w2, if GGGGG is a sufficiently large finite simple group, every element of GGGGG is a product of their word values. In fact, it is even possible [27] to choose subsets S1S1S_(1)S_{1}S1 and S2S2S_(2)S_{2}S2 of the sets of word values of w1w1w_(1)w_{1}w1 and w2w2w_(2)w_{2}w2 such that S1S2=GS1S2=GS_(1)S_(2)=GS_{1} S_{2}=GS1S2=G and |Si|=O(|G|1/2log1/2|G|)Si=O|G|1/2log1/2â¡|G||S_(i)|=O(|G|^(1//2)log^(1//2)|G|)\left|S_{i}\right|=O\left(|G|^{1 / 2} \log ^{1 / 2}|G|\right)|Si|=O(|G|1/2log1/2â¡|G|). The set of values of any word is a union of conjugacy classes, and the basic strategy of the proof is to try to find conjugacy classes C1C1C_(1)C_{1}C1 and C2C2C_(2)C_{2}C2 contained in the word values of w1w1w_(1)w_{1}w1 and w2w2w_(2)w_{2}w2, respectively, such that C1C2=GC1C2=GC_(1)C_(2)=GC_{1} C_{2}=GC1C2=G and very few elements of GGGGG have significantly fewer representations as such products than one would expect. Then a random choice of subsets Si⊂CiSi⊂CiS_(i)subC_(i)S_{i} \subset C_{i}Si⊂Ci of suitable size can almost always be slightly modified to work.
In general, the probability distribution on the word values of wwwww obtained by evaluation at a uniformly distributed random element of GdGdG^(d)G^{d}Gd is far from uniform. For instance, for g∈A3ng∈A3ng inA_(3n)g \in \mathrm{A}_{3 n}g∈A3n uniformly distributed, the probability that g3=1g3=1g^(3)=1g^{3}=1g3=1 is at least |A3n|−1A3n−1|A_(3n)|^(-1)\left|\mathrm{A}_{3 n}\right|^{-1}|A3n|−1 times the number of elements of shape 3n3n3^(n)3^{n}3n, i.e.,
for nnnnn sufficiently large. Thus, setting w1=w2=x3w1=w2=x3w_(1)=w_(2)=x^(3)w_{1}=w_{2}=x^{3}w1=w2=x3, the probability that the product of cubes of two randomly chosen elements is 1 is at least |A3n|−2/3−2/3nA3n−2/3−2/3n|A_(3n)|^(-2//3-2//3n)\left|A_{3 n}\right|^{-2 / 3-2 / 3 n}|A3n|−2/3−2/3n, which, for large nnnnn, makes the distribution far from uniform, at least in the L∞L∞L^(oo)L^{\infty}L∞ sense.
Using exponential character estimates, Shalev, Tiep, and I proved [25, theorem 4] that for any word wwwww, there exists kkkkk such that as |G|→∞|G|→∞|G|rarr oo|G| \rightarrow \infty|G|→∞, the L∞L∞L^(oo)L^{\infty}L∞-deviation from uniformity in the product of kkkkk independent randomly generated values of wwwww goes to 0 . The dependence of kkkkk on wwwww is unavoidable, as the above example suggests. On the other hand, the L1L1L^(1)L^{1}L1-deviation from uniformity goes to 0 in the product of two independent randomly generated values of wwwww, for any nontrivial word wwwww [25, THEOREM 1]. I do not know what to expect for LpLpL^(p)L^{p}Lp-deviation for 1<p<∞1<p<∞1 < p < oo1<p<\infty1<p<∞.
6. FUCHSIAN GROUPS
For g,m≥0g,m≥0g,m >= 0g, m \geq 0g,m≥0, let d1,…,dm≥2d1,…,dm≥2d_(1),dots,d_(m) >= 2d_{1}, \ldots, d_{m} \geq 2d1,…,dm≥2 be integers. For
Assume e<0e<0e < 0e<0e<0, so ΓΓGamma\GammaΓ is an oriented, cocompact Fuchsian group. Let GGGGG be a finite group, and let C1,…,CmC1,…,CmC_(1),dots,C_(m)C_{1}, \ldots, C_{m}C1,…,Cm denote conjugacy classes in GGGGG of elements whose orders divide d1,…,dmd1,…,dmd_(1),dots,d_(m)d_{1}, \ldots, d_{m}d1,…,dm,
respectively. The Frobenius formula can be regarded as the g=0g=0g=0g=0g=0 case of a more general formula for the number of homomorphisms Γ→GΓ→GGamma rarr G\Gamma \rightarrow GΓ→G mapping xixix_(i)x_{i}xi to an element of CiCiC_(i)C_{i}Ci for all iiiii,
In favorable situations, one can prove that the χ=1χ=1chi=1\chi=1χ=1 term dominates all the others combined, in which case one has a good estimate for the number of such homomorphisms. Using this, Liebeck and Shalev proved [32, THEOREM 1.5] that if g≥2g≥2g >= 2g \geq 2g≥2, and GGGGG is a simple of Lie type group of rank rrrrr, then
By the same method, employing the character bounds of [28], one obtains the same estimate whenever eeeee is less than some absolute constant, regardless of the value of ggggg. It would be interesting to know whether this is true in general for e<0e<0e < 0e<0e<0. Some evidence in favor of this idea is given in [21,33], but for small qqqqq the problem is open.
An interesting geometric consequence of the method of Liebeck-Shalev is that if G_G_G_\underline{G}G_ is a simple algebraic group of rank rrrrr and g≥2g≥2g >= 2g \geq 2g≥2, the morphism G_2g→G_G_2g→G_G_^(2g)rarrG_\underline{G}^{2 g} \rightarrow \underline{G}G_2g→G_ given by the word [y1,z1]⋯[yg,zg]y1,z1⋯yg,zg[y_(1),z_(1)]cdots[y_(g),z_(g)]\left[y_{1}, z_{1}\right] \cdots\left[y_{g}, z_{g}\right][y1,z1]⋯[yg,zg] has all fibers of the same dimension, (2g−1)dimG_(2g−1)dimâ¡G_(2g-1)dim G_(2 g-1) \operatorname{dim} \underline{G}(2g−1)dimâ¡G_. This has been refined by Avraham Aizenbud and Nir Avni, who proved [1] that for g≥373g≥373g >= 373g \geq 373g≥373, the fibers of this morphism are reduced and have rational singularities. It would be interesting to extend this to the case of general Fuchsian groups. For instance, does there exist an absolute constant kkk\mathrm{k}k such that for all simple algebraic groups G_G_G_\underline{G}G_ and conjugacy classes C_1,…,C_mC_1,…,C_mC__(1),dots,C__(m)\underline{C}_{1}, \ldots, \underline{C}_{m}C_1,…,C_m with dimC_1+dimâ¡C_1+dim C__(1)+\operatorname{dim} \underline{C}_{1}+dimâ¡C_1+⋯+dimC_m>kdimG_⋯+dimâ¡C_m>kdimâ¡G_cdots+dim C__(m) > k dim G_\cdots+\operatorname{dim} \underline{C}_{m}>k \operatorname{dim} \underline{G}⋯+dimâ¡C_m>kdimâ¡G_, all fibers of the multiplication morphism C_1×⋯×C_m→G_C_1×⋯×C_m→G_C__(1)xx cdots xxC__(m)rarrG_\underline{C}_{1} \times \cdots \times \underline{C}_{m} \rightarrow \underline{G}C_1×⋯×C_m→G_ are reduced with rational singularities The ideas of Glazer-Hendel [11,12] may be applicable.
For g=1g=1g=1g=1g=1, we can no longer hope for equidimensional fibers, since the generic fiber dimension is dimG_dimâ¡G_dim G_\operatorname{dim} \underline{G}dimâ¡G_, while the fiber over the identity element has dimension r+dimG_r+dimâ¡G_r+dim G_r+\operatorname{dim} \underline{G}r+dimâ¡G_. However, Zhipeng LuLuLu\mathrm{Lu}Lu and I proved [20] that for G_=SLnG_=SLnG_=SL_(n)\underline{G}=\mathrm{SL}_{n}G_=SLn, all fibers over noncentral elements have dimension G_G_G_\underline{G}G_. It would be interesting to know whether this is true for general simple algebraic groups G_G_G_\underline{G}G_.
FUNDING
This work was partially supported by the National Science Foundation.
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MICHAEL J. LARSEN
Department of Mathematics, Indiana University, Bloomington, IN 47405, U.S.A., mjlarsen@indiana.edu
FINITE APPROXIMATIONS AS A TOOL FOR STUDYING TRIANGULATED CATEGORIES
AMNON NEEMAN
Abstract
Small, finite entities are easier and simpler to manipulate than gigantic, infinite ones. Consequently, huge chunks of mathematics are devoted to methods reducing the study of big, cumbersome objects to an analysis of their finite building blocks. The manifestation of this general pattern, in the study of derived and triangulated categories, dates back almost to the beginnings of the subject-more precisely to articles by Illusie in SGA6, way back in the early 1970s.
What is new, at least new in the world of derived and triangulated categories, is that one gets extra mileage from analyzing more carefully and quantifying more precisely just how efficiently one can estimate infinite objects by finite ones. This leads one to the study of metrics on triangulated categories, and of how accurately an object can be approximated by finite objects of bounded size.
In every branch of mathematics, we try to solve complicated problems by reducing to simpler ones, and from antiquity people have used finite approximations to study infinite objects. Naturally, whenever a new field comes into being, one of the first developments is to try to understand what should be the right notion of finiteness in the discipline. Derived and triangulated categories were introduced by Verdier in his PhD thesis in the mid-1960s (although the published version only appeared much later in [38]). Not surprisingly, the idea of studying the finite objects in these categories followed suit soon after, see Illusie [13-15].
Right from the start there was a pervasive discomfort with derived and triangulated categories-the intuition that had been built up, in dealing with concrete categories, mostly fails for triangulated categories. In case the reader is wondering: in the previous sentence the word "concrete" has a precise, technical meaning, and it is an old theorem of Freyd [10,11] that triangulated categories often are not concrete. Further testimony, to the strangeness of derived and triangulated categories, is that it took two decades before the intuitive notion of finiteness, which dates back to Illusie's articles [13-15], was given its correct formal definition. The following may be found in [23, DEFINITION 1.1].
Definition 1.1. Let TTT\mathscr{T}T be a triangulated category with coproducts. An object C∈TC∈TC inTC \in \mathscr{T}C∈T is called compact if Hom(C,−)Homâ¡(C,−)Hom(C,-)\operatorname{Hom}(C,-)Homâ¡(C,−) commutes with coproducts. The full subcategory of all compact objects will be denoted by TcTcT^(c)\mathscr{T}^{c}Tc.
Remark 1.2. I have often been asked where the name "compact" came from. In the preprint version of [23], these objects went by a different name, but the (anonymous) referee did not like it. I was given a choice: I was allowed to baptize them either "compact" or "small."
Who was I to argue with a referee?
Once one has a good working definition of what the finite objects ought to be, the next step is to give the right criterion which guarantees that the category has "enough" of them. For triangulated categories, the right definition did not come until [24, DEFINITION 1.7].
Definition 1.3. Let TTT\mathscr{T}T be a triangulated category with coproducts. The category TTT\mathscr{T}T is called compactly generated if every nonzero object X∈TX∈TX inTX \in \mathscr{T}X∈T admits a nonzero map C→XC→XC rarr XC \rightarrow XC→X, with C∈TC∈TC inTC \in \mathscr{T}C∈T a compact object.
As the reader may have guessed from the name, compactly generated triangulated categories are those in which it is often possible to reduce general problems to questions about compact objects-which tend to be easier.
All of the above nowadays counts as "classical," meaning that it is two or more decades old and there is already a substantial and diverse literature exploiting the ideas. This article explores the recent developments that arose from trying to understand how efficiently one can approximate arbitrary objects by compact ones. We first survey the results obtained to date. This review is on the skimpy side, partly because there already are other, more expansive published accounts in the literature, but mostly because we want to leave ourselves space to suggest possible directions for future research. Thus the article can be
thought of as having two components: a bare-bone review of what has been achieved to date, occupying Sections 2 to 6 , followed by Section 7 which comprises suggestions of avenues that might merit further development.
Our review presents just enough detail so that the open questions, making up Section 7, can be formulated clearly and comprehensibly, and so that the significance and potential applications of the open questions can be illuminated. This has the unfortunate side effect that we give short shrift to the many deep, substantial contributions, made by numerous mathematicians, which preceded and inspired the work presented here. The author apologizes in advance for this omission, which is the inescapable corollary of page limits. The reader is referred to the other surveys of the subject, where more care is taken to attribute the ideas correctly to their originators, and give credit where credit is due.
We permit ourselves to gloss over difficult technicalities, nonchalantly skating by nuances and subtleties, with only an occasional passing reference to the other surveys or to the research papers for more detail.
The reader wishing to begin with examples and applications, to keep in mind through the forthcoming abstraction, is encouraged to first look at the Introduction to [31].
2. APPROXIMABLE TRIANGULATED CATEGORIES-THE FORMAL DEFINITION AS A VARIANT ON FOURIER SERIES
It is now time to start our review, offering a glimpse of the recent progress that was made by trying to measure how "complicated" an object is, in other words, how far it is from being compact. What follows is sufficiently new for there to be much room for improvement: the future will undoubtedly see cleaner, more elegant, and more general formulations. What is presented here is the current crude state of this emerging field.
Discussion 2.1. This section is devoted to defining approximable triangulated categories, and the definition is technical and at first sight could appear artificial, maybe even forbidding, It might help therefore to motivate it with an analogy.
Let S1S1S^(1)\mathbb{S}^{1}S1 be the circle, and let M(S1)MS1M(S^(1))M\left(\mathbb{S}^{1}\right)M(S1) be the set of all complex-valued, Lebesguemeasurable functions on S1S1S^(1)\mathbb{S}^{1}S1. As usual we view S1=R/ZS1=R/ZS^(1)=R//Z\mathbb{S}^{1}=\mathbb{R} / \mathbb{Z}S1=R/Z as the quotient of its universal cover RRR\mathbb{R}R by the fundamental group ZZZ\mathbb{Z}Z; this identifies functions on S1S1S^(1)\mathbb{S}^{1}S1 with periodic functions on RRR\mathbb{R}R with period 1. In particular the function g(x)=e2πixg(x)=e2Ï€ixg(x)=e^(2pi ix)g(x)=e^{2 \pi i x}g(x)=e2Ï€ix belongs to M(S1)MS1M(S^(1))M\left(\mathbb{S}^{1}\right)M(S1). And, for each ℓ∈Zℓ∈Zâ„“inZ\ell \in \mathbb{Z}ℓ∈Z, we have that g(x)ℓ=e2πiℓxg(x)â„“=e2Ï€iâ„“xg(x)^(â„“)=e^(2pi iâ„“x)g(x)^{\ell}=e^{2 \pi i \ell x}g(x)â„“=e2Ï€iâ„“x also belongs to M(S1)MS1M(S^(1))M\left(\mathbb{S}^{1}\right)M(S1). Given a norm on the space M(S1)MS1M(S^(1))M\left(\mathbb{S}^{1}\right)M(S1), for example, the LpLpL^(p)L^{p}Lp-norm, we can try to approximate arbitrary f∈M(S1)f∈MS1f in M(S^(1))f \in M\left(\mathbb{S}^{1}\right)f∈M(S1) by Laurent polynomials in ggggg, that is, look for complex numbers {λℓ∈C∣−n≤ℓ≤n}λℓ∈C∣−n≤ℓ≤n{lambda_(â„“)inC∣-n <= â„“ <= n}\left\{\lambda_{\ell} \in \mathbb{C} \mid-n \leq \ell \leq n\right\}{λℓ∈C∣−n≤ℓ≤n} such that
with ε>0ε>0epsi > 0\varepsilon>0ε>0 small. This leads us to the familiar territory of Fourier series.
Now imagine trying to do the same, but replacing M(S1)MS1M(S^(1))M\left(\mathbb{S}^{1}\right)M(S1) by a triangulated category. Given a triangulated category TTT\mathscr{T}T, which we assume to have coproducts, we would like to pretend to do Fourier analysis on it. We would need to choose:
(1) Some analog of the function g(x)=e2πixg(x)=e2Ï€ixg(x)=e^(2pi ix)g(x)=e^{2 \pi i x}g(x)=e2Ï€ix. Our replacement for this will be to choose a compact generator G∈TG∈TG inTG \in \mathscr{T}G∈T. Recall that a compact generator is a compact object G∈TG∈TG inTG \in \mathscr{T}G∈T such that every nonzero object X∈TX∈TX inTX \in \mathscr{T}X∈T admits a nonzero map G[i]→XG[i]→XG[i]rarr XG[i] \rightarrow XG[i]→X for some i∈Zi∈Zi inZi \in \mathbb{Z}i∈Z.
(2) We need to choose something like a metric, the analog of the LpLpL^(p)L^{p}Lp-norm on M(S1)MS1M(S^(1))M\left(\mathbb{S}^{1}\right)M(S1). For us this will be done by picking a t-structure (T≤0,T≥0)T≤0,T≥0(T <= 0,T^( >= 0))\left(\mathscr{T} \leq 0, \mathscr{T}^{\geq 0}\right)(T≤0,T≥0) on TTT\mathscr{T}T. The heuristic is that we will view a morphism E→FE→FE rarr FE \rightarrow FE→F in TTT\mathscr{T}T as "short" if, in the triangle E→F→DE→F→DE rarr F rarr DE \rightarrow F \rightarrow DE→F→D, the object DDDDD belongs to T≤−nT≤−nT^( <= -n)\mathscr{T}^{\leq-n}T≤−n for large nnnnn. We will come back to this in Discussion 6.10.
(3) We need to have an analog of the construction that passes, from the function g(x)=e2πixg(x)=e2Ï€ixg(x)=e^(2pi ix)g(x)=e^{2 \pi i x}g(x)=e2Ï€ix and the integer n>0n>0n > 0n>0n>0, to the vector space of trigonometric Laurent polynomials ∑ℓ=−nnλℓe2πiℓx∑ℓ=−nn λℓe2Ï€iâ„“xsum_(â„“=-n)^(n)lambda_(â„“)e^(2pi iâ„“x)\sum_{\ell=-n}^{n} \lambda_{\ell} e^{2 \pi i \ell x}∑ℓ=−nnλℓe2Ï€iâ„“x.
As it happens our solution to (3) is technical. We need a recipe that begins with the object GGGGG and the integer n>0n>0n > 0n>0n>0, and proceeds to cook up a collection of more objects. We ask the reader to accept it as a black box, with only a sketchy explanation just before Remark 2.3.
Black Box 2.2. Let TTT\mathscr{T}T be a triangulated category and let G∈TG∈TG inTG \in \mathscr{T}G∈T be an object. Let n>0n>0n > 0n>0n>0 be an integer. We will have occasion to refer to the following four full subcategories of TTT\mathscr{T}T :
To do this, let us build up the octahedron on the composable morphisms F→F→F rarrF \rightarrowF→D→D′D→D′D rarrD^(')D \rightarrow D^{\prime}D→D′. We end up with a diagram where the rows and columns are triangles
Combining Remark 2.5 with Facts 2.6 allows us to improve approximations through iteration. Hence part (2) of the definition below becomes natural, it iterates to provide arbitrarily good approximations.
Definition 2.7. Let TTT\mathscr{T}T be a triangulated category with coproducts. It is approximable if there exist a t-structure (T≤0,T≥0)(T≤0,T≥0)(T <= 0,T >= 0)(\mathscr{T} \leq 0, \mathscr{T} \geq 0)(T≤0,T≥0), a compact generator G∈TG∈TG inTG \in \mathscr{T}G∈T, and an integer n>0n>0n > 0n>0n>0 such that
(1) GGGGG belongs to T≤nT≤nT <= n\mathscr{T} \leq nT≤n and Hom(G,T≤−n)=0Homâ¡(G,T≤−n)=0Hom(G,T <= -n)=0\operatorname{Hom}(G, \mathscr{T} \leq-n)=0Homâ¡(G,T≤−n)=0;
Remark 2.8. While part (2) of Definition 2.7 comes motivated by the analogy with Fourier analysis, part (1) of the definition seems random. It requires the ttttt-structure, which is our replacement for the LpLpL^(p)L^{p}Lp-norm, to be compatible with the compact generator, which is the analog of g(x)=e2πixg(x)=e2Ï€ixg(x)=e^(2pi ix)g(x)=e^{2 \pi i x}g(x)=e2Ï€ix. As the reader will see in Proposition 5.5, this has the effect of uniquely specifying the ttttt-structure (up to equivalence). So maybe a better parallel would be to fix our norm to be a particularly nice one, for example, the L2L2L^(2)L^{2}L2-norm on M(S1)MS1M(S^(1))M\left(\mathbb{S}^{1}\right)M(S1).
Let me repeat myself: as with all new mathematics, Definition 2.7 should be viewed as provisional. In the remainder of this survey, we will discuss the applications as they now
stand, to highlight the power of the methods. But I would not be surprised in the slightest if future applications turn out to require modifications, and/or generalizations, of the definitions and of the theorems that have worked so far.
3. EXAMPLES OF APPROXIMABLE TRIANGULATED CATEGORIES
In Section 1 we gave the definition of approximable triangulated categories. The definition combines old, classical ingredients (t-structures and compact generators) with a new
theory is nonempty: we need to produce examples, categories people care about which satisfy the definition of approximability. The current section is devoted to the known examples of approximable triangulated categories. We repeat what we have said before: the subject is in its infancy, there could well be many more examples out there.
Example 3.1. Let TTT\mathscr{T}T be a triangulated category with coproducts. If G∈TG∈TG inTG \in \mathscr{T}G∈T is a compact generator such that Hom(G,G[i])=0Homâ¡(G,G[i])=0Hom(G,G[i])=0\operatorname{Hom}(G, G[i])=0Homâ¡(G,G[i])=0 for all i>0i>0i > 0i>0i>0, then the category TTT\mathscr{T}T is approximable.
This example turns out to be easy, the reader is referred to [29, EXAMPLE 3.3] for the (short) proof. Special cases include
(1) T=D(R−Mod)T=D(R−Mod)T=D(R-Mod)\mathscr{T}=\mathbf{D}(R-\mathrm{Mod})T=D(R−Mod), where RRRRR is a dga with Hi(R)=0Hi(R)=0H^(i)(R)=0H^{i}(R)=0Hi(R)=0 for i>0i>0i > 0i>0i>0;
(2) The homotopy category of spectra.
Example 3.2. If XXXXX is a quasicompact, separated scheme, then the category Dqc(X)Dqc(X)D_(qc)(X)\mathbf{D}_{\mathbf{q c}}(X)Dqc(X) is approximable. We remind the reader of the traditional notation being used here: the category D(X)D(X)D(X)\mathbf{D}(X)D(X) is the unbounded derived category of complexes of sheaves of OXOXO_(X)\mathscr{O}_{X}OX-modules, and the full subcategory Dqc(X)⊂D(X)Dqc(X)⊂D(X)D_(qc)(X)subD(X)\mathbf{D}_{\mathbf{q c}}(X) \subset \mathbf{D}(X)Dqc(X)⊂D(X) has for objects the complexes with quasicoherent cohomology.
The proof of the approximability of Dqc(X)Dqc(X)D_(qc)(X)\mathbf{D}_{\mathbf{q c}}(X)Dqc(X) is not trivial. The category has a standard ttttt-structure, that part is easy. The existence of a compact generator GGGGG needs proof, it may be found in Bondal and Van den Bergh [6, THEOREM 3.1.1(II)]. Their proof is not constructive, it is only an existence proof, but it does give enough information to deduce that part (1) of Definition 2.7 is satisfied by every compact generator (indeed, it is satisfied by every compact object). See [6, THEOREM 3.1.1(I)]. But it is a challenge to show that we may choose a compact generator GGGGG and an integer n>0n>0n > 0n>0n>0 in such a way that Definition 2.7(2) is satisfied.
If we further assume that XXXXX is of finite type over a noetherian ring RRRRR, then the (relatively intricate) proof of the approximability of Dqc(X)Dqc(X)D_(qc)(X)\mathbf{D}_{\mathbf{q c}}(X)Dqc(X) occupies [33, SECTIONS 4 AND 5]. The little trick, that extends the result to all quasicompact and separated XXXXX, was not observed until later: it appears in [29, LEMMA 3.5].
Example 3.3. It is a theorem that, under mild hypotheses, the recollement of any two approximable triangulated categories is approximable. To state the "mild hypotheses" precisely: suppose we are given a recollement of triangulated categories
with RRR\mathscr{R}R and TTT\mathscr{T}T approximable. Assume further that the category SSS\mathscr{S}S is compactly generated, and any compact object H∈SH∈SH inSH \in \mathscr{S}H∈S has the property that Hom(H,H[i])=0Homâ¡(H,H[i])=0Hom(H,H[i])=0\operatorname{Hom}(H, H[i])=0Homâ¡(H,H[i])=0 for i≫0i≫0i≫0i \gg 0i≫0. Then the category SSS\mathscr{S}S is also approximable.
The reader can find the proof in [7, THEOREM 4.1], it is the main result in the paper. The bulk of the article is devoted to developing the machinery necessary to prove the theoremhence it is worth noting that this machinery has since demonstrated usefulness in other contexts, see the subsequent articles [27,28].
There is a beautiful theory of noncommutative schemes, and a rich literature studying them. And many of the interesting examples of such schemes are obtained as recollements of ordinary schemes, or of admissible pieces of them. Thus the theorem that recollements of approximable triangulated categories are approximable gives a wealth of new examples of approximable triangulated categories.
Since this ICM is being held in St. Petersburg, it would be remiss not to mention that the theory of noncommutative algebraic geometry, in the sense of the previous paragraph, is a subject to which Russian mathematicians have contributed a vast amount. The seminal work of Bondal, Kontsevich, Kuznetsov, Lunts, and Orlov immediately springs to mind. For a beautiful introduction to the field, the reader might wish to look at the early sections of Orlov [34]. The later sections prove an amazing new theorem, but the early ones give a lovely survey of the background. In fact, the theory sketched in this survey was born when I was trying to read and understand Orlov's beautiful article.
4. APPLICATIONS: STRONG GENERATION
We begin by reminding the reader of a classical definition, going back to Bondal and Van den Bergh [6].
The first application of approximability is the proof of the following two theorems.
Theorem 4.2. Let XXXXX be a quasicompact, separated scheme. The derived category of perfect complexes on XXXXX, denoted here by Dperf (X)Dperf (X)D^("perf ")(X)\mathbf{D}^{\text {perf }}(X)Dperf (X), is regular if and only if XXXXX has a cover by open subsets Spec(Ri)⊂XSpecâ¡Ri⊂XSpec(R_(i))sub X\operatorname{Spec}\left(R_{i}\right) \subset XSpecâ¡(Ri)⊂X, with each RiRiR_(i)R_{i}Ri of finite global dimension.
Remark 4.3. If XXXXX is noetherian and separated, then Theorem 4.2 specializes to saying that Dperf (X)Dperf (X)D^("perf ")(X)\mathbf{D}^{\text {perf }}(X)Dperf (X) is regular if and only if XXXXX is regular and finite-dimensional. Hence the terminology.
Theorem 4.4. Let XXXXX be a noetherian, separated, finite-dimensional, quasiexcellent scheme. Then the category Db(Coh(X))Db(Cohâ¡(X))D^(b)(Coh(X))\mathbf{D}^{b}(\operatorname{Coh}(X))Db(Cohâ¡(X)), the bounded derived category of coherent sheaves on XXXXX, is always regular.
Remark 4.5. The reader is referred to [33] and to Aoki [4] for the proofs of Theorems 4.2 and 4.4. More precisely, for Theorem 4.2 see [33, THEOREM 0.5]. About Theorem 4.4: if we add the assumption that every closed subvariety of XXXXX admits a regular alteration then the result may be found in [33, THEOREM 0.15], but Aoki [4] found a lovely argument that allowed him to extend the statement to all quasiexcellent XXXXX.
There is a rich literature on strong generation, with beautiful papers by many authors. In the introduction to [33], as well as in [26] and [31, sEction 7], the reader can find an extensive discussion of (some of) this fascinating work and of the way Theorems 4.2 and 4.4 compare to the older literature. For a survey taking an entirely different tack, see Minami [22], which places in historical perspective a couple of the key steps in the proofs that [33] gives for Theorems 4.2 and 4.4.
Since all of this is now well documented in the published literature, let us focus the remainder of the current survey on the other applications of approximability. Those are all still in preprint form, see [27-29], although there are (published) surveys in [31, SECTIONs 8 AND 9] and in [30]. Those surveys are fuller and more complete than the sketchy one we are about to embark on. As we present the material, we will feel free to refer the reader to the more extensive surveys whenever we deem it appropriate.
5. THE FREEDOM IN THE CHOICE OF COMPACT GENERATOR AND T-STRUCTURE
Definition 2.7 tells us that a triangulated category TTT\mathscr{T}T with coproducts is approximable if there exist, in TTT\mathscr{T}T, a compact generator GGGGG and a t-structure (T≤0,T≥0)(T≤0,T≥0)(T <= 0,T >= 0)(\mathscr{T} \leq 0, \mathscr{T} \geq 0)(T≤0,T≥0) satisfying some properties. The time has come to explore just how free we are in the choice of the compact generator and of the ttttt-structure. To address this question we begin by formulating:
Definition 5.1. Let TTT\mathscr{T}T be a triangulated category. Then two t-structures (T1≤0,T1≥0)T1≤0,T1≥0(T_(1)^( <= 0),T_(1)^( >= 0))\left(\mathscr{T}_{1}^{\leq 0}, \mathscr{T}_{1}^{\geq 0}\right)(T1≤0,T1≥0) and (T2≤0,T2≥0)T2≤0,T2≥0(T_(2)^( <= 0),T_(2)^( >= 0))\left(\mathscr{T}_{2}^{\leq 0}, \mathscr{T}_{2}^{\geq 0}\right)(T2≤0,T2≥0) are declared equivalent if there exists an integer n>0n>0n > 0n>0n>0 such that
Lemma 5.3. If GGGGG and HHHHH are two compact generators for the triangulated category TTT\mathscr{T}T, then the two ttttt-structures (TG≤0,TG≥0)TG≤0,TG≥0(T_(G)^( <= 0),T_(G)^( >= 0))\left(\mathscr{T}_{G}^{\leq 0}, \mathscr{T}_{G}^{\geq 0}\right)(TG≤0,TG≥0) and (TH≤0,TH≥0)TH≤0,TH≥0(T_(H)^( <= 0),T_(H)^( >= 0))\left(\mathscr{T}_{H}^{\leq 0}, \mathscr{T}_{H}^{\geq 0}\right)(TH≤0,TH≥0) are equivalent as in Definition 5.2.
As it happens, the proof of Lemma 5.3 is easy, the interested reader can find it in [29, Remark 0.15]. And Lemma 5.3 leads us to:
Definition 5.4. Let TTT\mathscr{T}T be a triangulated category in which there exists a compact generator. We define the preferred equivalence class of ttttt-structures as follows: a ttttt-structure belongs to the preferred equivalence class if it is equivalent to (TG≤0,TG≥0)TG≤0,TG≥0(T_(G)^( <= 0),T_(G)^( >= 0))\left(\mathscr{T}_{G}^{\leq 0}, \mathscr{T}_{G}^{\geq 0}\right)(TG≤0,TG≥0) for some compact generator G∈TG∈TG inTG \in \mathscr{T}G∈T, and by Lemma 5.3 it is equivalent to (TH≤0,TH≥0)TH≤0,TH≥0(T_(H)^( <= 0),T_(H)^( >= 0))\left(\mathscr{T}_{H}^{\leq 0}, \mathscr{T}_{H}^{\geq 0}\right)(TH≤0,TH≥0) for every compact generator HHHHH.
The following is also not too hard, and may be found in [29, PROPOSITIONS 2.4 AND 2.6].
Proposition 5.5. Let TTT\mathscr{T}T be an approximable triangulated category. Then for any ttttt-structure (T≤0,T≥0)T≤0,T≥0(T <= 0,T^( >= 0))\left(\mathscr{T} \leq 0, \mathscr{T}^{\geq 0}\right)(T≤0,T≥0) in the preferred equivalence class, and for any compact generator H∈TH∈TH inTH \in \mathscr{T}H∈T, there exists an integer n>0n>0n > 0n>0n>0 (which may depend on HHHHH and on the ttttt-structure), satisfying
(1) HHHHH belongs to T≤nT≤nT <= n\mathscr{T} \leq nT≤n and Hom(H,T≤−n)=0Homâ¡(H,T≤−n)=0Hom(H,T <= -n)=0\operatorname{Hom}(H, \mathscr{T} \leq-n)=0Homâ¡(H,T≤−n)=0;
Moreover, if HHHHH is a compact generator, (T≤0,T≥0)(T≤0,T≥0)(T <= 0,T >= 0)(\mathscr{T} \leq 0, \mathscr{T} \geq 0)(T≤0,T≥0) is a ttttt-structure, and there exists an integer n>0n>0n > 0n>0n>0 satisfying (1) and (2) above, then the ttttt-structure (T≤0,T≥0)(T≤0,T≥0)(T <= 0,T >= 0)(\mathscr{T} \leq 0, \mathscr{T} \geq 0)(T≤0,T≥0) must belong to the preferred equivalence class.
Remark 5.6. Strangely enough, the value of Proposition 5.5 can be that it allows us to find an explicit t-structure in the preferred equivalence class.
However, the combination of [33, THEOREM 5.8] and [29, LEMMA 3.5] tells us that the category Dqc(X)Dqc(X)D_(qc)(X)\mathbf{D}_{\mathbf{q c}}(X)Dqc(X) is approximable, and it so happens that the ttt\mathrm{t}t-structure used in the proof, that is, the ttt\mathrm{t}t-structure for which a compact generator HHHHH and an integer n>0n>0n > 0n>0n>0 satisfying (1) and (2) of Proposition 5.5 are shown to exist, happens to be the standard t-structure. From Proposition 5.5, we now deduce that the standard ttttt-structure is in the preferred equivalence class.
6. STRUCTURE THEOREMS IN APPROXIMABLE TRIANGULATED CATEGORIES
An approximable triangulated category TTT\mathscr{T}T must have a compact generator GGGGG, and Definition 5.4 constructed for us a preferred equivalence class of ttttt-structures-namely all
those equivalent to (TG≤0,TG≥0)TG≤0,TG≥0(T_(G)^( <= 0),T_(G)^( >= 0))\left(\mathscr{T}_{G}^{\leq 0}, \mathscr{T}_{G}^{\geq 0}\right)(TG≤0,TG≥0). Recall that, for any t-structure (T≤0,T≥0)T≤0,T≥0(T <= 0,T^( >= 0))\left(\mathscr{T} \leq 0, \mathscr{T}^{\geq 0}\right)(T≤0,T≥0), it is customary to define
It is an easy exercise to show, directly from Definition 5.1, that equivalent t-structures give rise to identical T−,T+T−,T+T^(-),T^(+)\mathscr{T}^{-}, \mathscr{T}^{+}T−,T+, and TbTbT^(b)\mathscr{T}^{b}Tb. Therefore triangulated categories with a single compact generator, and in particular approximable triangulated categories, have preferred subcategories T−,T+T−,T+T^(-),T^(+)\mathscr{T}^{-}, \mathscr{T}^{+}T−,T+, and TbTbT^(b)\mathscr{T}^{b}Tb, which are intrinsic-they are simply those corresponding to any ttt\mathrm{t}t-structure in the preferred equivalence class. In the remainder of this survey, we will assume that T−,T+T−,T+T^(-),T^(+)\mathscr{T}^{-}, \mathscr{T}^{+}T−,T+, and TbTbT^(b)\mathscr{T}^{b}Tb always stand for the preferred ones.
In the heuristics of Discussion 2.1(2), we told the reader that a t-structure (T≤0,T≥0)T≤0,T≥0(T <= 0,T^( >= 0))\left(\mathscr{T} \leq 0, \mathscr{T}^{\geq 0}\right)(T≤0,T≥0) is to be viewed as a metric on TTT\mathscr{T}T. In Definition 6.1 below, the heuristic is that we construct a full subcategory Tc−Tc−T_(c)^(-)\mathscr{T}_{c}^{-}Tc−to be the closure of TcTcT^(c)\mathscr{T}^{c}Tc with respect to any of the (equivalent) metrics that come from t-structures in the preferred equivalence class.
Definition 6.1. Let TTT\mathscr{T}T be an approximable triangulated category. The full subcategory Tc−Tc−T_(c)^(-)\mathscr{T}_{c}^{-}Tc− is given by
Ob(Tc−)={F∈T| For every integer n>0 and for every t-structure (T≤0,T≥0) in the preferred equivalence class, there exists an exact triangle E→F→D in T with E∈Tc and D∈T≤−n}Obâ¡Tc−=F∈T For every integer n>0 and for every t-structure (T≤0,T≥0) in the preferred equivalence class,  there exists an exact triangle E→F→D in T with E∈Tc and D∈T≤−nOb(T_(c)^(-))={F inT|[" For every integer "n > 0" and for every t-structure "],[(T <= 0","T >= 0)" in the preferred equivalence class, "],[" there exists an exact triangle "E rarr F rarr D" in "T],[" with "E inT^(c)" and "D inT <= -n]}\operatorname{Ob}\left(\mathscr{T}_{c}^{-}\right)=\left\{F \in \mathscr{T} \left\lvert\, \begin{array}{c}
\text { For every integer } n>0 \text { and for every t-structure } \\
(\mathscr{T} \leq 0, \mathscr{T} \geq 0) \text { in the preferred equivalence class, } \\
\text { there exists an exact triangle } E \rightarrow F \rightarrow D \text { in } \mathscr{T} \\
\text { with } E \in \mathscr{T}^{c} \text { and } D \in \mathscr{T} \leq-n
\end{array}\right.\right\}Obâ¡(Tc−)={F∈T| For every integer n>0 and for every t-structure (T≤0,T≥0) in the preferred equivalence class,  there exists an exact triangle E→F→D in T with E∈Tc and D∈T≤−n}
Remark 6.2. Let TTT\mathscr{T}T be an approximable triangulated category. Aside from the classical, full subcategory TcTcT^(c)\mathscr{T}^{c}Tc of compact objects, which we encountered back in Definition 1.1, we have in this section concocted five more intrinsic, full subcategories of TTT\mathscr{T}T : they are T−T−T^(-)\mathscr{T}^{-}T−, T+,Tb,Tc−T+,Tb,Tc−T^(+),T^(b),T_(c)^(-)\mathscr{T}^{+}, \mathscr{T}^{b}, \mathscr{T}_{c}^{-}T+,Tb,Tc−, and TcbTcbT_(c)^(b)\mathscr{T}_{c}^{b}Tcb. It can be proved that all six subcategories, that is, the old TcTcT^(c)\mathscr{T}^{c}Tc and the five new ones, are thick subcategories of TTT\mathscr{T}T. In particular, each of them is a triangulated category.
Example 6.3. It becomes interesting to figure out what all these categories come down to in examples.
Let XXXXX be a quasicompact, separated scheme. From Example 3.2, we know that the category T=Dqc(X)T=Dqc(X)T=D_(qc)(X)\mathscr{T}=\mathbf{D}_{\mathbf{q c}}(X)T=Dqc(X) is approximable, and in Remark 5.6 we noted that the standard ttt\mathrm{t}t-structure is in the preferred equivalence class. This can be used to show that, for T=Dqc(X)T=Dqc(X)T=D_(qc)(X)\mathscr{T}=\mathbf{D}_{\mathbf{q c}}(X)T=Dqc(X), we have
where the last two equalities assume that the scheme XXXXX is noetherian, and all six categories on the right of the equalities have their traditional meanings.
The reader can find an extensive discussion of the claims above in [31], more precisely in the paragraphs between [31, PROPOSITION 8.10] and [31, THEOREM 8.16]. That discussion
goes beyond the scope of the current survey, it analyzes the categories Tcb⊂Tc−Tcb⊂Tc−T_(c)^(b)subT_(c)^(-)\mathscr{T}_{c}^{b} \subset \mathscr{T}_{c}^{-}Tcb⊂Tc−in the generality of non-noetherian schemes, where they still have a classical description-of course, not involving the category of coherent sheaves. After all coherent sheaves do not behave well for non-noetherian schemes.
Remark 6.4. In this survey we spent some effort introducing the notion of approximable triangulated categories. In Example 3.2 we told the reader that it is a theorem (and not a trivial one) that, as long as a scheme XXXXX is quasicompact and separated, the derived category Dqc(X)Dqc(X)D_(qc)(X)\mathbf{D}_{\mathbf{q c}}(X)Dqc(X) is approximable. In this section we showed that every approximable triangulated category comes with canonically defined, intrinsic subcategories T−,T+,Tb,Tc,Tc−T−,T+,Tb,Tc,Tc−T^(-),T^(+),T^(b),T^(c),T_(c)^(-)\mathscr{T}^{-}, \mathscr{T}^{+}, \mathscr{T}^{b}, \mathscr{T}^{c}, \mathscr{T}_{c}^{-}T−,T+,Tb,Tc,Tc−, and TcbTcbT_(c)^(b)\mathscr{T}_{c}^{b}Tcb, and in Example 6.3 we informed the reader that, in the special case where T=Dqc(X)T=Dqc(X)T=D_(qc)(X)\mathscr{T}=\mathbf{D}_{\mathbf{q c}}(X)T=Dqc(X), these turn out to be Dqc−(X),Dqc+(X),Dqcb(X),Dperf (X),Dcoh −(X)Dqc−(X),Dqc+(X),Dqcb(X),Dperf (X),Dcoh −(X)D_(qc)^(-)(X),D_(qc)^(+)(X),D_(qc)^(b)(X),D^("perf ")(X),D_("coh ")^(-)(X)\mathbf{D}_{\mathbf{q c}}^{-}(X), \mathbf{D}_{\mathbf{q c}}^{+}(X), \mathbf{D}_{\mathbf{q c}}^{b}(X), \mathbf{D}^{\text {perf }}(X), \mathbf{D}_{\text {coh }}^{-}(X)Dqc−(X),Dqc+(X),Dqcb(X),Dperf (X),Dcoh −(X), and Dcoh b(X)Dcoh b(X)D_("coh ")^(b)(X)\mathbf{D}_{\text {coh }}^{b}(X)Dcoh b(X), respectively.
Big deal. This teaches us that the traditional subcategories Dqc−(X),Dqc+(X),Dqcb(X)Dqc−(X),Dqc+(X),Dqcb(X)D_(qc)^(-)(X),D_(qc)^(+)(X),D_(qc)^(b)(X)\mathbf{D}_{\mathbf{q c}}^{-}(X), \mathbf{D}_{\mathbf{q c}}^{+}(X), \mathbf{D}_{\mathbf{q c}}^{b}(X)Dqc−(X),Dqc+(X),Dqcb(X), Dperf (X),Dcoh −(X)Dperf (X),Dcoh −(X)D^("perf ")(X),D_("coh ")^(-)(X)\mathbf{D}^{\text {perf }}(X), \mathbf{D}_{\text {coh }}^{-}(X)Dperf (X),Dcoh −(X), and Dcoh b(X)Dcoh b(X)D_("coh ")^(b)(X)\mathbf{D}_{\text {coh }}^{b}(X)Dcoh b(X) of the category Dqc(X)Dqc(X)D_(qc)(X)\mathbf{D}_{\mathbf{q c}}(X)Dqc(X) all have intrinsic descriptions. This might pass as a curiosity, unless we can actually use it to prove something we care about that we did not use to know.
Discussion 6.5. To motivate the next theorem, it might help to think of the parallel with functional analysis.
Let M(R)M(R)M(R)M(\mathbb{R})M(R) be the vector space of Lebesgue-measurable, real-valued functions on RRR\mathbb{R}R. Given any two functions f,g∈M(R)f,g∈M(R)f,g in M(R)f, g \in M(\mathbb{R})f,g∈M(R), we can pair them by integrating the product, that is, we form the pairing
which turns out to be an isometry of Banach spaces.
The category-theoretic version is that on any category TTT\mathscr{T}T there is the pairing sending two objects A,B∈TA,B∈TA,B inTA, B \in \mathscr{T}A,B∈T to Hom(A,B)Homâ¡(A,B)Hom(A,B)\operatorname{Hom}(A, B)Homâ¡(A,B). Of course, this pairing is not symmetric, we have to keep track of the position of AAAAA and of BBBBB in Hom(A,B)Homâ¡(A,B)Hom(A,B)\operatorname{Hom}(A, B)Homâ¡(A,B). If RRRRR is a commutative ring and TTT\mathscr{T}T happens to be an RRRRR-linear category, then Hom(A,B)Homâ¡(A,B)Hom(A,B)\operatorname{Hom}(A, B)Homâ¡(A,B) is an RRRRR-module and the pairing delivers a map
where the op keeps track of the variable in the first position. And now we are free to restrict to subcategories of TTT\mathscr{T}T.
If TTT\mathscr{T}T happens to be approximable and RRRRR-linear, we have just learned that it comes with six intrinsic subcategories T−,T+,Tb,Tc,Tc−T−,T+,Tb,Tc,Tc−T^(-),T^(+),T^(b),T^(c),T_(c)^(-)\mathscr{T}^{-}, \mathscr{T}^{+}, \mathscr{T}^{b}, \mathscr{T}^{c}, \mathscr{T}_{c}^{-}T−,T+,Tb,Tc,Tc−, and TcbTcbT_(c)^(b)\mathscr{T}_{c}^{b}Tcb. We are free to restrict the Hom pairing to any couple of them. This gives us 36 possible pairings, and each of those yields two maps from a subcategory to the dual of another. There are 72 cases we could study, and the theorem below tells us something useful about four of those.
Theorem 6.6. Let RRRRR be a noetherian ring, and let TTT\mathscr{T}T be an RRRRR-linear, approximable triangulated category. Suppose there exists in TTT\mathscr{T}T a compact generator GGGGG so that Hom(G,G[n])Homâ¡(G,G[n])Hom(G,G[n])\operatorname{Hom}(G, G[n])Homâ¡(G,G[n]) is a finite RRRRR-module for all n∈Zn∈Zn inZn \in \mathbb{Z}n∈Z. Consider the two functors
defined by the formulas Y(B)=Hom(−,B)Y(B)=Homâ¡(−,B)Y(B)=Hom(-,B)\mathscr{Y}(B)=\operatorname{Hom}(-, B)Y(B)=Homâ¡(−,B) and Y~(A)=Hom(A,−)Y~(A)=Homâ¡(A,−)widetilde(Y)(A)=Hom(A,-)\widetilde{\mathscr{Y}}(A)=\operatorname{Hom}(A,-)Y~(A)=Homâ¡(A,−), as in Discussion 6.5 . Now consider the following composites:
(1) The functor YYY\mathscr{Y}Y is full, and the essential image consists of the locally finite homological functors (see Explanation 6.7 for the definition of locally finite functors). The composite YYY\mathscr{Y}Y â—‹ iiiii is fully faithful, and the essential image consists of the finite homological functors (again, see Explanation 6.7 for the definition).
Explanation 6.7. In the statement of Theorem 6.6, the locally finite functors, either of the form H:[Tc]op →RH:Tcop →RH:[T^(c)]^("op ")rarr RH:\left[\mathscr{T}^{c}\right]^{\text {op }} \rightarrow RH:[Tc]op →R-Mod or of the form H:Tcb→RH:Tcb→RH:T_(c)^(b)rarr RH: \mathscr{T}_{c}^{b} \rightarrow RH:Tcb→R-Mod, are the functors such that
(1) H(A[i])H(A[i])H(A[i])H(A[i])H(A[i]) is a finite RRRRR-module for every i∈Zi∈Zi inZi \in \mathbb{Z}i∈Z and every AAAAA in either TcTcT^(c)\mathscr{T}^{c}Tc or TcbTcbT_(c)^(b)\mathscr{T}_{c}^{b}